[{"Name":"The Indefinite Integral","TopicPlaylistFirstVideoID":0,"Duration":null,"Videos":[{"Watched":false,"Name":"Introduction","Duration":"8m 35s","ChapterTopicVideoID":1518,"CourseChapterTopicPlaylistID":82571,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.com/Images/Videos_Thumbnails/1518.jpeg","UploadDate":"2019-11-14T01:31:16.5170000","DurationForVideoObject":"PT8M35S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:05.220","Text":"In this clip, I\u0027ll be introducing a concept called the indefinite integral."},{"Start":"00:05.220 ","End":"00:07.875","Text":"I\u0027d like to introduce it in the form of a game."},{"Start":"00:07.875 ","End":"00:14.010","Text":"Up till now, the game is been where I give you a function and you find its derivative,"},{"Start":"00:14.010 ","End":"00:15.975","Text":"a process called differentiation."},{"Start":"00:15.975 ","End":"00:18.240","Text":"This time we\u0027re going to reverse the game,"},{"Start":"00:18.240 ","End":"00:20.790","Text":"a kind of anti-differentiation,"},{"Start":"00:20.790 ","End":"00:23.445","Text":"where I give you the derivative,"},{"Start":"00:23.445 ","End":"00:25.395","Text":"for example, I\u0027ll say to you,"},{"Start":"00:25.395 ","End":"00:31.710","Text":"I have a function and I\u0027m telling you that it\u0027s derivative f prime of x is 2x."},{"Start":"00:31.710 ","End":"00:35.520","Text":"I want you to tell me what the original function was,"},{"Start":"00:35.520 ","End":"00:41.225","Text":"f of x is what in order that its derivative be 2x?"},{"Start":"00:41.225 ","End":"00:43.400","Text":"In this case, you\u0027re probably familiar enough."},{"Start":"00:43.400 ","End":"00:46.355","Text":"You\u0027ve done enough differentiation to say right away,"},{"Start":"00:46.355 ","End":"00:48.020","Text":"\"Yes, not a problem."},{"Start":"00:48.020 ","End":"00:52.235","Text":"I\u0027ll give you f of x is equal to x squared,\" and"},{"Start":"00:52.235 ","End":"00:56.855","Text":"you\u0027d be right because the derivative of x squared is indeed 2x."},{"Start":"00:56.855 ","End":"00:59.390","Text":"But how about if I said to you,"},{"Start":"00:59.390 ","End":"01:03.770","Text":"f of x is equal to x squared minus 4,"},{"Start":"01:03.770 ","End":"01:07.040","Text":"you\u0027d say, \"If I differentiate this function,"},{"Start":"01:07.040 ","End":"01:08.734","Text":"I also get 2x.\""},{"Start":"01:08.734 ","End":"01:14.540","Text":"That\u0027s not all. How about f of x equals x squared plus 100?"},{"Start":"01:14.540 ","End":"01:17.330","Text":"Then again you say, \"If I differentiate this,"},{"Start":"01:17.330 ","End":"01:19.160","Text":"yes, I do get 2x.\""},{"Start":"01:19.160 ","End":"01:21.575","Text":"There seems to be more than one answer."},{"Start":"01:21.575 ","End":"01:25.550","Text":"In fact, if I call this f1 of x, f2 of x,"},{"Start":"01:25.550 ","End":"01:30.900","Text":"f3 of x, they all have the property that the derivative is 2x."},{"Start":"01:30.900 ","End":"01:33.165","Text":"They\u0027re all antiderivative."},{"Start":"01:33.165 ","End":"01:37.010","Text":"It seems to be a rule here that if I, in general,"},{"Start":"01:37.010 ","End":"01:44.400","Text":"would take a function f of x equals x squared and put any c here, c for constant."},{"Start":"01:44.400 ","End":"01:47.565","Text":"C could be minus 100, it could be 0,"},{"Start":"01:47.565 ","End":"01:50.215","Text":"it could be 1 1/2, or anything,"},{"Start":"01:50.215 ","End":"01:54.350","Text":"such a function when differentiated would give me 2x."},{"Start":"01:54.350 ","End":"01:58.565","Text":"This in fact is the general form of the answer to the question,"},{"Start":"01:58.565 ","End":"02:01.955","Text":"what\u0027s the function whose derivative is 2x?"},{"Start":"02:01.955 ","End":"02:04.700","Text":"There\u0027s name for this game we\u0027ve been playing."},{"Start":"02:04.700 ","End":"02:07.855","Text":"This game is called integration."},{"Start":"02:07.855 ","End":"02:11.135","Text":"This is called an indefinite integral."},{"Start":"02:11.135 ","End":"02:13.910","Text":"The word indefinite because it\u0027s not a definite one,"},{"Start":"02:13.910 ","End":"02:17.390","Text":"like this is a definite function whose derivative is 2x,"},{"Start":"02:17.390 ","End":"02:22.470","Text":"but this got a constant in it, unspecified or indefinite."},{"Start":"02:22.470 ","End":"02:24.260","Text":"This is the indefinite integral,"},{"Start":"02:24.260 ","End":"02:27.155","Text":"and going from here to here is called integration."},{"Start":"02:27.155 ","End":"02:31.130","Text":"I\u0027d like to introduce some concepts and notation."},{"Start":"02:31.130 ","End":"02:38.210","Text":"A function like x squared whose derivative is 2x is called a primitive of 2x."},{"Start":"02:38.210 ","End":"02:39.830","Text":"That\u0027s just a mathematical word."},{"Start":"02:39.830 ","End":"02:46.255","Text":"I would write here that this x squared is a primitive of 2x."},{"Start":"02:46.255 ","End":"02:48.635","Text":"I\u0027ve heard the term antiderivative,"},{"Start":"02:48.635 ","End":"02:50.540","Text":"but we won\u0027t be using that here."},{"Start":"02:50.540 ","End":"02:52.295","Text":"We\u0027ll be using the word primitive."},{"Start":"02:52.295 ","End":"02:55.720","Text":"Likewise, x squared minus 4 is also a primitive of 2x."},{"Start":"02:55.720 ","End":"02:56.930","Text":"I don\u0027t want to write it again,"},{"Start":"02:56.930 ","End":"02:58.535","Text":"so I\u0027ll write ditto."},{"Start":"02:58.535 ","End":"03:03.845","Text":"X squared plus 100 is also a primitive of 2x, so ditto."},{"Start":"03:03.845 ","End":"03:06.695","Text":"But this one, which is in the general form,"},{"Start":"03:06.695 ","End":"03:12.545","Text":"is actually the indefinite integral of 2x."},{"Start":"03:12.545 ","End":"03:15.470","Text":"I also have to write in mathematical terms."},{"Start":"03:15.470 ","End":"03:17.915","Text":"There\u0027s a way of writing this down."},{"Start":"03:17.915 ","End":"03:19.790","Text":"You won\u0027t have to say those words each time and"},{"Start":"03:19.790 ","End":"03:22.595","Text":"people in other languages will know what you mean."},{"Start":"03:22.595 ","End":"03:26.175","Text":"What we do, we start with the 2x."},{"Start":"03:26.175 ","End":"03:29.915","Text":"This is the one that we have to find the general primitive of."},{"Start":"03:29.915 ","End":"03:31.865","Text":"We write the 2x, and before it,"},{"Start":"03:31.865 ","End":"03:37.280","Text":"we write an elongated S. This is actually an old-fashioned S. After it,"},{"Start":"03:37.280 ","End":"03:39.875","Text":"to close the brackets,"},{"Start":"03:39.875 ","End":"03:48.430","Text":"we write dx and we write the answer that this is equal to x squared plus c. Again,"},{"Start":"03:48.430 ","End":"03:51.725","Text":"to give a mathematical notation to the game which says,"},{"Start":"03:51.725 ","End":"03:53.630","Text":"given the function 2x,"},{"Start":"03:53.630 ","End":"03:56.390","Text":"find the general function whose derivative is 2x,"},{"Start":"03:56.390 ","End":"03:57.680","Text":"we write it as follows."},{"Start":"03:57.680 ","End":"04:00.410","Text":"This is like open brackets, close brackets."},{"Start":"04:00.410 ","End":"04:02.645","Text":"We start with the elongated S,"},{"Start":"04:02.645 ","End":"04:04.325","Text":"which is the integration sign,"},{"Start":"04:04.325 ","End":"04:06.765","Text":"and we end with dx."},{"Start":"04:06.765 ","End":"04:09.170","Text":"There are historical reasons for why it\u0027s dx,"},{"Start":"04:09.170 ","End":"04:10.715","Text":"but I don\u0027t want to get into that."},{"Start":"04:10.715 ","End":"04:13.265","Text":"Here we write the general primitive,"},{"Start":"04:13.265 ","End":"04:16.640","Text":"the general function whose derivative is 2x."},{"Start":"04:16.640 ","End":"04:20.720","Text":"Now let me add some labels and we\u0027ll say that"},{"Start":"04:20.720 ","End":"04:25.770","Text":"this whole thing is the indefinite integral of 2x."},{"Start":"04:25.770 ","End":"04:29.045","Text":"This piece here is called, as we know,"},{"Start":"04:29.045 ","End":"04:31.720","Text":"a primitive of 2x,"},{"Start":"04:31.720 ","End":"04:35.330","Text":"and this is a general constant."},{"Start":"04:35.330 ","End":"04:38.600","Text":"That\u0027s how the indefinite integral is built."},{"Start":"04:38.600 ","End":"04:43.370","Text":"It\u0027s built of a specific primitive of 2x in this case,"},{"Start":"04:43.370 ","End":"04:45.530","Text":"plus the general constant."},{"Start":"04:45.530 ","End":"04:47.510","Text":"You don\u0027t just write 7 here or something,"},{"Start":"04:47.510 ","End":"04:49.280","Text":"you leave it as the letter c,"},{"Start":"04:49.280 ","End":"04:51.215","Text":"c is the accepted letter,"},{"Start":"04:51.215 ","End":"04:53.870","Text":"and that gives us the indefinite integral."},{"Start":"04:53.870 ","End":"04:56.450","Text":"I\u0027ll give another example to make it clear."},{"Start":"04:56.450 ","End":"05:02.080","Text":"This time I want the most general function whose derivative is 7."},{"Start":"05:02.080 ","End":"05:07.400","Text":"What I do is I indicate this by writing the integral of 7,"},{"Start":"05:07.400 ","End":"05:09.799","Text":"7 as a constant function,"},{"Start":"05:09.799 ","End":"05:12.110","Text":"and then dx to close"},{"Start":"05:12.110 ","End":"05:18.280","Text":"the elongated S. Then I think which function differentiates to give 7,"},{"Start":"05:18.280 ","End":"05:19.720","Text":"one answer is 7x,"},{"Start":"05:19.720 ","End":"05:22.260","Text":"so 7x is a primitive."},{"Start":"05:22.260 ","End":"05:24.785","Text":"Then I add a general constant."},{"Start":"05:24.785 ","End":"05:27.935","Text":"This is the indefinite integral of 7."},{"Start":"05:27.935 ","End":"05:31.220","Text":"The one right here is usually the simpler one."},{"Start":"05:31.220 ","End":"05:34.550","Text":"If I see here these 3,"},{"Start":"05:34.550 ","End":"05:36.320","Text":"then the x squared looks the simplest."},{"Start":"05:36.320 ","End":"05:39.170","Text":"But it wouldn\u0027t really matter if you used any one of them because if I"},{"Start":"05:39.170 ","End":"05:42.020","Text":"write 100 plus a constant or just a constant,"},{"Start":"05:42.020 ","End":"05:44.015","Text":"it\u0027s still just the general constant."},{"Start":"05:44.015 ","End":"05:47.780","Text":"The good thing about this is that you can check yourself if you\u0027ve"},{"Start":"05:47.780 ","End":"05:51.380","Text":"got the right indefinite integral because all you have to do is differentiate,"},{"Start":"05:51.380 ","End":"05:54.560","Text":"and if you know to differentiate 7x and gets a 7,"},{"Start":"05:54.560 ","End":"05:58.610","Text":"there\u0027s a way of checking yourself to see if you\u0027ve got the indefinite integral right."},{"Start":"05:58.610 ","End":"06:00.350","Text":"Let\u0027s go for another example."},{"Start":"06:00.350 ","End":"06:06.205","Text":"Let\u0027s take the integral of 4x cubed,"},{"Start":"06:06.205 ","End":"06:09.185","Text":"dx just closes the integral sign."},{"Start":"06:09.185 ","End":"06:13.400","Text":"Now, there\u0027s no method but we can make a guess because we\u0027re familiar,"},{"Start":"06:13.400 ","End":"06:15.305","Text":"we\u0027ve done a lot of differentiating."},{"Start":"06:15.305 ","End":"06:18.185","Text":"You might recall that if we took x to the 4th,"},{"Start":"06:18.185 ","End":"06:19.370","Text":"then we differentiate it,"},{"Start":"06:19.370 ","End":"06:21.335","Text":"we get 4x cubed."},{"Start":"06:21.335 ","End":"06:23.260","Text":"That would be a primitive."},{"Start":"06:23.260 ","End":"06:27.020","Text":"If we add the general constant,"},{"Start":"06:27.020 ","End":"06:30.275","Text":"then we have the indefinite integral of 4x cubed,"},{"Start":"06:30.275 ","End":"06:31.945","Text":"which is x to the 4th."},{"Start":"06:31.945 ","End":"06:34.580","Text":"As before, we can always check ourselves."},{"Start":"06:34.580 ","End":"06:39.845","Text":"Derivative of x to the 4th is 4x cubed and derivative of constant is nothing."},{"Start":"06:39.845 ","End":"06:42.395","Text":"Let\u0027s just go for one last example."},{"Start":"06:42.395 ","End":"06:51.005","Text":"What is the integral of 1 over twice the square root of x, and dx."},{"Start":"06:51.005 ","End":"06:54.740","Text":"In other words, what I\u0027m asking in game terms is,"},{"Start":"06:54.740 ","End":"06:57.875","Text":"find me a function that if I differentiate it,"},{"Start":"06:57.875 ","End":"07:00.230","Text":"I\u0027ll get 1 over twice square root of x,"},{"Start":"07:00.230 ","End":"07:03.725","Text":"or its derivative is 1 over twice square root of x."},{"Start":"07:03.725 ","End":"07:06.380","Text":"This is one of those you\u0027re supposed to have memorized,"},{"Start":"07:06.380 ","End":"07:09.545","Text":"and the answer is just the square root of x."},{"Start":"07:09.545 ","End":"07:11.795","Text":"We\u0027ve done this enough to memorize that"},{"Start":"07:11.795 ","End":"07:15.905","Text":"the derivative of square root of x is 1 over twice the square root of x."},{"Start":"07:15.905 ","End":"07:17.770","Text":"This is a primitive."},{"Start":"07:17.770 ","End":"07:20.700","Text":"If I add the general constant c,"},{"Start":"07:20.700 ","End":"07:23.475","Text":"then I get the indefinite integral."},{"Start":"07:23.475 ","End":"07:26.990","Text":"We check ourselves, differentiate this, we get this."},{"Start":"07:26.990 ","End":"07:29.450","Text":"That\u0027s the last example."},{"Start":"07:29.450 ","End":"07:34.220","Text":"But notice that up till now we\u0027ve basically been doing guesswork."},{"Start":"07:34.220 ","End":"07:37.250","Text":"We\u0027re starting with things that we\u0027re familiar with;"},{"Start":"07:37.250 ","End":"07:40.790","Text":"the derivative of linear functions of a power of x,"},{"Start":"07:40.790 ","End":"07:42.260","Text":"of the square root of x."},{"Start":"07:42.260 ","End":"07:43.715","Text":"These are all familiar."},{"Start":"07:43.715 ","End":"07:47.765","Text":"What would happen if tomorrow someone would say to you,"},{"Start":"07:47.765 ","End":"07:56.735","Text":"I want the integral of 1 plus x plus x squared over x dx."},{"Start":"07:56.735 ","End":"07:58.640","Text":"How would you go about it then?"},{"Start":"07:58.640 ","End":"08:03.860","Text":"Well, just as we had rules for differentiation and we had a whole load of rules,"},{"Start":"08:03.860 ","End":"08:06.955","Text":"we\u0027re going to have rules for integration,"},{"Start":"08:06.955 ","End":"08:09.945","Text":"for finding the integrals of functions."},{"Start":"08:09.945 ","End":"08:12.930","Text":"That\u0027s coming up, but not in this clip."},{"Start":"08:12.930 ","End":"08:15.470","Text":"Today I can\u0027t give you the answer."},{"Start":"08:15.470 ","End":"08:19.625","Text":"I can just say question mark because I don\u0027t have all these rules and I can\u0027t"},{"Start":"08:19.625 ","End":"08:24.005","Text":"just guess a function which if I differentiate it would give me this."},{"Start":"08:24.005 ","End":"08:26.900","Text":"You might be really lucky and make a very good guess,"},{"Start":"08:26.900 ","End":"08:28.865","Text":"but we really need the system."},{"Start":"08:28.865 ","End":"08:33.110","Text":"These integration rules will becoming up probably in the next clip."},{"Start":"08:33.110 ","End":"08:35.670","Text":"For now, that\u0027s it."}],"ID":1532},{"Watched":false,"Name":"Integration Rules - Part A","Duration":"15m 3s","ChapterTopicVideoID":1519,"CourseChapterTopicPlaylistID":82571,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.com/Images/Videos_Thumbnails/1519.jpeg","UploadDate":"2019-11-14T01:31:35.2300000","DurationForVideoObject":"PT15M3S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:05.720","Text":"In this clip, I\u0027m going to be teaching some of the rules for doing integration."},{"Start":"00:05.720 ","End":"00:10.185","Text":"This is in the context of indefinite integrals which we, hopefully,"},{"Start":"00:10.185 ","End":"00:14.700","Text":"you\u0027ve studied already, perhaps in the clip right before this."},{"Start":"00:14.700 ","End":"00:22.780","Text":"An indefinite integral as a primitive or antiderivative in the general sense."},{"Start":"00:24.650 ","End":"00:28.365","Text":"I\u0027m assuming you know what the indefinite integral is."},{"Start":"00:28.365 ","End":"00:30.480","Text":"Let\u0027s start with rule number 1."},{"Start":"00:30.480 ","End":"00:34.875","Text":"Rule number 1, we want to know how to integrate a constant,"},{"Start":"00:34.875 ","End":"00:40.300","Text":"the integral of a constant, say a dx."},{"Start":"00:40.300 ","End":"00:43.655","Text":"Before I give you the rule, let\u0027s see if we can figure it out ourselves."},{"Start":"00:43.655 ","End":"00:45.125","Text":"If I had, say,"},{"Start":"00:45.125 ","End":"00:48.140","Text":"the integral of 4dx,"},{"Start":"00:48.140 ","End":"00:49.625","Text":"what am I asking for?"},{"Start":"00:49.625 ","End":"00:54.200","Text":"I\u0027m asking for the general function whose derivative is 4."},{"Start":"00:54.200 ","End":"00:58.235","Text":"Now a function whose derivative is 4 should be fairly easy."},{"Start":"00:58.235 ","End":"01:03.520","Text":"If you\u0027ve been awake till now, you would say 4x."},{"Start":"01:03.520 ","End":"01:10.705","Text":"Of course, it could also be 4x plus 3 or 4x plus 100 or 4x minus 2."},{"Start":"01:10.705 ","End":"01:13.835","Text":"All these when derived, would give you 4."},{"Start":"01:13.835 ","End":"01:16.790","Text":"If I put 4x plus C here,"},{"Start":"01:16.790 ","End":"01:21.935","Text":"that\u0027s the general primitive function for 4."},{"Start":"01:21.935 ","End":"01:26.150","Text":"Likewise, when we have the integral of adx,"},{"Start":"01:26.150 ","End":"01:29.285","Text":"It\u0027s just instead of full we have a,"},{"Start":"01:29.285 ","End":"01:32.425","Text":"so we\u0027d expect it to be ax."},{"Start":"01:32.425 ","End":"01:39.650","Text":"As always, plus C. Let\u0027s check that that works."},{"Start":"01:39.650 ","End":"01:43.760","Text":"For example, if I have the integral of just"},{"Start":"01:43.760 ","End":"01:50.000","Text":"1dx and use this rule and it\u0027s a 1x plus C. In other words,"},{"Start":"01:50.000 ","End":"01:55.025","Text":"x plus C. We can always check ourselves, differentiate this."},{"Start":"01:55.025 ","End":"01:58.520","Text":"The derivative of x is 1 and C doesn\u0027t give anything."},{"Start":"01:58.520 ","End":"02:00.920","Text":"That seems to be right."},{"Start":"02:00.920 ","End":"02:03.680","Text":"That\u0027s it. That\u0027s rule number 1."},{"Start":"02:03.680 ","End":"02:06.300","Text":"Now rule number 2."},{"Start":"02:06.380 ","End":"02:11.030","Text":"Rule number 2 is the integral of x to the n,"},{"Start":"02:11.030 ","End":"02:13.040","Text":"where n is some whole number."},{"Start":"02:13.040 ","End":"02:15.920","Text":"This is the most useful of all the rules,"},{"Start":"02:15.920 ","End":"02:19.500","Text":"probably the most used certainly."},{"Start":"02:21.860 ","End":"02:24.470","Text":"Let\u0027s try an example."},{"Start":"02:24.470 ","End":"02:30.845","Text":"Let\u0027s just try the integral of x to the 4dx."},{"Start":"02:30.845 ","End":"02:35.630","Text":"Not so clear what function differentiated gives x to the 4th."},{"Start":"02:35.630 ","End":"02:41.505","Text":"You\u0027re probably thinking something like x to the 5th, but not precisely."},{"Start":"02:41.505 ","End":"02:43.610","Text":"We actually do need a rule here."},{"Start":"02:43.610 ","End":"02:53.300","Text":"And the rule is that it\u0027s equal to x to the power of n plus 1 over n plus 1."},{"Start":"02:53.300 ","End":"03:00.310","Text":"In other words, you increase the power by 1 and divide by that new power."},{"Start":"03:00.310 ","End":"03:04.625","Text":"As usual, we add plus C at the end."},{"Start":"03:04.625 ","End":"03:06.995","Text":"Let\u0027s see, let\u0027s try it out."},{"Start":"03:06.995 ","End":"03:10.445","Text":"What happens here. Here, n is 4."},{"Start":"03:10.445 ","End":"03:17.040","Text":"We get x^4 plus 1 is 5."},{"Start":"03:17.040 ","End":"03:27.215","Text":"Then that 5, we divide by it over 5 plus C. We can always check by differentiating."},{"Start":"03:27.215 ","End":"03:32.065","Text":"The derivative of x to the 5th is 5x^4th."},{"Start":"03:32.065 ","End":"03:34.400","Text":"The 5 just stays there."},{"Start":"03:34.400 ","End":"03:36.545","Text":"It\u0027s a constant that\u0027s multiplying,"},{"Start":"03:36.545 ","End":"03:38.975","Text":"or it\u0027s 1/5 of x^1/5."},{"Start":"03:38.975 ","End":"03:43.745","Text":"The 1/5 and the 5 just cancel and we\u0027re left with x^4th."},{"Start":"03:43.745 ","End":"03:45.665","Text":"The rule seems to work,"},{"Start":"03:45.665 ","End":"03:49.340","Text":"but this is important warning."},{"Start":"03:49.340 ","End":"03:51.575","Text":"We can\u0027t divide by 0,"},{"Start":"03:51.575 ","End":"03:56.925","Text":"so we must not have n being minus 1 and"},{"Start":"03:56.925 ","End":"04:02.990","Text":"n is not equal to minus 1 will have a separate rule just for the minus 1 case."},{"Start":"04:02.990 ","End":"04:07.740","Text":"This is important and is not minus 1."},{"Start":"04:07.900 ","End":"04:12.005","Text":"In other words, if it was 1 over x,"},{"Start":"04:12.005 ","End":"04:14.315","Text":"which is x to the minus 1,"},{"Start":"04:14.315 ","End":"04:16.040","Text":"we couldn\u0027t do it with this rule."},{"Start":"04:16.040 ","End":"04:20.070","Text":"So 1 over x doesn\u0027t work with this."},{"Start":"04:21.820 ","End":"04:24.740","Text":"Let\u0027s try a few more examples."},{"Start":"04:24.740 ","End":"04:28.250","Text":"Since it is, since this is such an important rule,"},{"Start":"04:28.250 ","End":"04:30.785","Text":"what should we try next."},{"Start":"04:30.785 ","End":"04:37.190","Text":"Let\u0027s try integral of x dx."},{"Start":"04:37.190 ","End":"04:40.390","Text":"Now very often we need to do a bit of manipulating here."},{"Start":"04:40.390 ","End":"04:41.920","Text":"It\u0027s only very slight,"},{"Start":"04:41.920 ","End":"04:44.950","Text":"but sometimes we have to take things from the denominator to"},{"Start":"04:44.950 ","End":"04:50.050","Text":"the numerator and play with the exponents and square roots and so on."},{"Start":"04:50.050 ","End":"04:53.830","Text":"Here, for example, it\u0027s almost like this."},{"Start":"04:53.830 ","End":"04:55.465","Text":"If I just put 1 here,"},{"Start":"04:55.465 ","End":"05:00.335","Text":"then it falls under the x^n category, haven\u0027t changed anything."},{"Start":"05:00.335 ","End":"05:05.320","Text":"This becomes then x to the power of increase by 1,"},{"Start":"05:05.320 ","End":"05:09.365","Text":"so I get 2 and I divide by the new exponent,"},{"Start":"05:09.365 ","End":"05:11.985","Text":"which is 2 here also."},{"Start":"05:11.985 ","End":"05:15.940","Text":"I almost don\u0027t say the plus C, just write it."},{"Start":"05:16.250 ","End":"05:19.510","Text":"At the end we check ourselves."},{"Start":"05:19.510 ","End":"05:22.250","Text":"Properly check it here, but let\u0027s do it here."},{"Start":"05:22.250 ","End":"05:26.450","Text":"If we have 1/2x squared,"},{"Start":"05:26.450 ","End":"05:29.455","Text":"which is what this is, and differentiate it."},{"Start":"05:29.455 ","End":"05:32.329","Text":"Let\u0027s say we differentiate it prime,"},{"Start":"05:32.329 ","End":"05:36.110","Text":"then we get 1/2 times 2, which is 1."},{"Start":"05:36.110 ","End":"05:38.615","Text":"We\u0027ll get 1/2 times 2."},{"Start":"05:38.615 ","End":"05:42.700","Text":"Here x^2 minus 1,"},{"Start":"05:42.700 ","End":"05:44.745","Text":"which is just 1."},{"Start":"05:44.745 ","End":"05:47.270","Text":"We basically get back to the original."},{"Start":"05:47.270 ","End":"05:51.530","Text":"We have this option of checking our answer for"},{"Start":"05:51.530 ","End":"05:54.050","Text":"an integration exercise by just differentiating"},{"Start":"05:54.050 ","End":"05:58.650","Text":"the answer and seeing if we get back to where we came from."},{"Start":"06:01.160 ","End":"06:04.690","Text":"I don\u0027t want this clutter here."},{"Start":"06:04.690 ","End":"06:07.320","Text":"How about another example?"},{"Start":"06:07.320 ","End":"06:13.350","Text":"In fact, we\u0027ll do quite few more examples since this is such an important rule."},{"Start":"06:14.590 ","End":"06:24.155","Text":"The next one I\u0027ll do will be the integral of 1 over x squared dx."},{"Start":"06:24.155 ","End":"06:29.135","Text":"Now again, I said it\u0027s not always in the form x^n."},{"Start":"06:29.135 ","End":"06:31.630","Text":"Here, it\u0027s 1 over x^n."},{"Start":"06:31.630 ","End":"06:34.595","Text":"But if we remember our exponent rules,"},{"Start":"06:34.595 ","End":"06:39.920","Text":"this is equal to the integral of x^negative 2."},{"Start":"06:39.920 ","End":"06:41.089","Text":"When it\u0027s in the denominator,"},{"Start":"06:41.089 ","End":"06:46.040","Text":"you can put it in the numerator if you reverse the sign dx."},{"Start":"06:46.040 ","End":"06:55.415","Text":"Now we can apply the rule here and say we increase this power by 1,"},{"Start":"06:55.415 ","End":"06:57.395","Text":"so we get minus 1,"},{"Start":"06:57.395 ","End":"06:59.590","Text":"so it\u0027s x to the minus 1."},{"Start":"06:59.590 ","End":"07:01.625","Text":"Then we divide by that."},{"Start":"07:01.625 ","End":"07:10.325","Text":"Increased power over minus 1 and plus C. Could leave it like that."},{"Start":"07:10.325 ","End":"07:15.020","Text":"But usually if the original exercise was without exponents,"},{"Start":"07:15.020 ","End":"07:19.830","Text":"but using the denominator the answer should also be that way."},{"Start":"07:19.830 ","End":"07:23.690","Text":"I would write it as dx minus 1 as 1 over x."},{"Start":"07:23.690 ","End":"07:25.415","Text":"The minus can go on top,"},{"Start":"07:25.415 ","End":"07:30.330","Text":"so we get minus 1 over x here plus C."},{"Start":"07:31.770 ","End":"07:37.130","Text":"Another example will be,"},{"Start":"07:37.170 ","End":"07:40.255","Text":"I just pushed this stuff over here,"},{"Start":"07:40.255 ","End":"07:42.085","Text":"so we\u0027ll have a bit more space."},{"Start":"07:42.085 ","End":"07:48.040","Text":"What I want to do now is try the integral of square root of x dx."},{"Start":"07:48.040 ","End":"07:51.040","Text":"Again, this doesn\u0027t look like x to the n,"},{"Start":"07:51.040 ","End":"07:55.180","Text":"but it can be made to look like it if we just use the fact"},{"Start":"07:55.180 ","End":"08:00.370","Text":"that the square root of something is that something to the power of a half."},{"Start":"08:00.370 ","End":"08:05.950","Text":"I have x to the power of 1/2 dx, sorry."},{"Start":"08:05.950 ","End":"08:09.700","Text":"Now it does look like this."},{"Start":"08:09.700 ","End":"08:14.995","Text":"We can use the rule and say that this is equal to x to the power of"},{"Start":"08:14.995 ","End":"08:22.930","Text":"1/2 plus 1 is 1.5 or 3/2 whatever,"},{"Start":"08:22.930 ","End":"08:29.200","Text":"and divide by 3 over 2 plus C."},{"Start":"08:29.200 ","End":"08:38.170","Text":"We can leave it like this or you can convert it to using square roots."},{"Start":"08:38.170 ","End":"08:41.560","Text":"I\u0027ll just do that quickly without too much explanation."},{"Start":"08:41.560 ","End":"08:47.170","Text":"What we can say is dividing by 3 over 2 is like multiplying by 2/3."},{"Start":"08:47.170 ","End":"08:54.190","Text":"X to the 1.5 is x times x to the 1/2 or we can just say the square root of x cubed."},{"Start":"08:54.190 ","End":"08:59.200","Text":"That\u0027s maybe a better way to do it if this square root of x cubed,"},{"Start":"08:59.200 ","End":"09:03.260","Text":"but still we\u0027ll plus C at the end."},{"Start":"09:03.450 ","End":"09:10.690","Text":"Another example, let\u0027s see something like integral of 1"},{"Start":"09:10.690 ","End":"09:19.645","Text":"over x times the 4th root of x dx."},{"Start":"09:19.645 ","End":"09:22.944","Text":"Certainly doesn\u0027t look like this much anymore."},{"Start":"09:22.944 ","End":"09:26.270","Text":"But we do a little bit of algebra first."},{"Start":"09:28.710 ","End":"09:37.030","Text":"Look x is x to the power of 1 and the 4th root of x is x to the power of 1/4."},{"Start":"09:37.030 ","End":"09:42.100","Text":"What we have is 1 over x to the power of 1 and 1/4."},{"Start":"09:42.100 ","End":"09:45.220","Text":"I could have written 5/4,"},{"Start":"09:45.220 ","End":"09:50.500","Text":"I just chose this way, dx."},{"Start":"09:50.610 ","End":"10:00.820","Text":"Now, I can also put it in the numerator and get rid of this 1 over,"},{"Start":"10:00.820 ","End":"10:02.649","Text":"so I can get the integral."},{"Start":"10:02.649 ","End":"10:12.160","Text":"If I make this negative of x to the power of minus 1 and 1/4 dx."},{"Start":"10:12.160 ","End":"10:17.470","Text":"Now I can take the integral by using this formula because now it"},{"Start":"10:17.470 ","End":"10:22.495","Text":"is of the form x to the power of n. This is x to the power of,"},{"Start":"10:22.495 ","End":"10:31.765","Text":"now n plus 1 is minus 1 and 1/4 plus 1 is just minus 1/4 over,"},{"Start":"10:31.765 ","End":"10:37.250","Text":"I have to put this over the new power, over minus 1/4."},{"Start":"10:37.650 ","End":"10:44.500","Text":"All in all, what I can do if I plus C of course."},{"Start":"10:44.500 ","End":"10:49.270","Text":"If I simplify this and I won\u0027t go into any great detail because this is the answer."},{"Start":"10:49.270 ","End":"10:54.470","Text":"The minus 1/4 comes to be minus 4 at the top."},{"Start":"10:55.050 ","End":"11:00.460","Text":"X to the minus 1/4 is like the 4th root of x,"},{"Start":"11:00.460 ","End":"11:04.165","Text":"but also in the denominator because of the minus."},{"Start":"11:04.165 ","End":"11:11.215","Text":"It\u0027s minus 4 over the fourth root of x and"},{"Start":"11:11.215 ","End":"11:19.790","Text":"plus C. That\u0027s a simplified answer though this will certainly do."},{"Start":"11:21.030 ","End":"11:26.620","Text":"I think that\u0027s enough examples for rule number 2."},{"Start":"11:26.620 ","End":"11:34.280","Text":"What I\u0027m going to do is just clean up a bit there."},{"Start":"11:34.470 ","End":"11:38.515","Text":"I think we can move back down again."},{"Start":"11:38.515 ","End":"11:41.470","Text":"I want to get onto rule number 3."},{"Start":"11:41.470 ","End":"11:45.115","Text":"Rule number 3 relates to rule number 2,"},{"Start":"11:45.115 ","End":"11:46.240","Text":"it answers that question,"},{"Start":"11:46.240 ","End":"11:50.305","Text":"what happens if n is equal to minus 1?"},{"Start":"11:50.305 ","End":"11:55.420","Text":"We want to know what is the integral of x to the minus 1"},{"Start":"11:55.420 ","End":"12:01.465","Text":"or what is the integral of 1 over x dx."},{"Start":"12:01.465 ","End":"12:05.530","Text":"But I want to go over again why we can\u0027t use rule number 2."},{"Start":"12:05.530 ","End":"12:09.040","Text":"Suppose we naively, we\u0027re going to use rule number 2."},{"Start":"12:09.040 ","End":"12:12.850","Text":"Then we would start off by writing this equals x to"},{"Start":"12:12.850 ","End":"12:17.470","Text":"the minus 1 dx and then say, rule number 2."},{"Start":"12:17.470 ","End":"12:22.270","Text":"We just have to increase the exponent, the power by 1,"},{"Start":"12:22.270 ","End":"12:26.980","Text":"and we get x to the 0 over 0 plus C."},{"Start":"12:26.980 ","End":"12:31.075","Text":"Then we see what nonsense we\u0027ve written because you can\u0027t divide by 0."},{"Start":"12:31.075 ","End":"12:33.010","Text":"This whole approach doesn\u0027t work."},{"Start":"12:33.010 ","End":"12:37.555","Text":"Let me erase it. Now, what is the answer?"},{"Start":"12:37.555 ","End":"12:44.670","Text":"You might remember that the derivative of the natural log of x is 1 over x."},{"Start":"12:44.670 ","End":"12:50.385","Text":"It seems like the integral of this is natural log"},{"Start":"12:50.385 ","End":"12:57.885","Text":"of x and plus C. Certainly if you differentiate this,"},{"Start":"12:57.885 ","End":"12:59.520","Text":"you will get this."},{"Start":"12:59.520 ","End":"13:04.170","Text":"But there\u0027s a fine technical point here that we have to get into."},{"Start":"13:04.170 ","End":"13:12.175","Text":"Notice that the domain of 1 over x is all x except 0."},{"Start":"13:12.175 ","End":"13:16.600","Text":"In other words, the domain here is x not equal to 0,"},{"Start":"13:16.600 ","End":"13:22.090","Text":"whereas the domain of the natural log is only the positive x."},{"Start":"13:22.090 ","End":"13:25.915","Text":"Here we have positive or negative in both cases 0 is disallowed,"},{"Start":"13:25.915 ","End":"13:28.314","Text":"but here we also have the negative."},{"Start":"13:28.314 ","End":"13:37.255","Text":"It turns out that what fixes this is if we put the absolute value around the x."},{"Start":"13:37.255 ","End":"13:41.275","Text":"Not only is this defined also for positive and negative,"},{"Start":"13:41.275 ","End":"13:43.810","Text":"but it even turns out to be the right answer if you"},{"Start":"13:43.810 ","End":"13:48.220","Text":"differentiate natural log of x when x is negative."},{"Start":"13:48.220 ","End":"13:50.935","Text":"I\u0027ll just show you briefly."},{"Start":"13:50.935 ","End":"13:53.200","Text":"It\u0027s not mandatory to follow,"},{"Start":"13:53.200 ","End":"13:55.630","Text":"I\u0027m just giving you for those who are curious,"},{"Start":"13:55.630 ","End":"13:57.565","Text":"if x is negative,"},{"Start":"13:57.565 ","End":"14:07.750","Text":"then natural log of absolute value of x is equal to natural log of minus x."},{"Start":"14:07.750 ","End":"14:09.685","Text":"That\u0027s what it is when x is negative."},{"Start":"14:09.685 ","End":"14:12.280","Text":"If we differentiate this,"},{"Start":"14:12.280 ","End":"14:17.770","Text":"what we get is the derivative of natural log is 1 over,"},{"Start":"14:17.770 ","End":"14:19.930","Text":"so it\u0027s 1 over minus x,"},{"Start":"14:19.930 ","End":"14:22.630","Text":"but times the internal derivative from the chain rule,"},{"Start":"14:22.630 ","End":"14:24.175","Text":"which is minus 1."},{"Start":"14:24.175 ","End":"14:26.935","Text":"It just comes out to be 1 over x also."},{"Start":"14:26.935 ","End":"14:28.750","Text":"Anyway that\u0027s briefly the reason,"},{"Start":"14:28.750 ","End":"14:32.875","Text":"formula it works and it\u0027s also defined in the same domains."},{"Start":"14:32.875 ","End":"14:35.740","Text":"Now I\u0027m going to erase this and you can forget it."},{"Start":"14:35.740 ","End":"14:38.995","Text":"You just have to remember to put"},{"Start":"14:38.995 ","End":"14:44.420","Text":"these bars around the x when we integrate 1 over x, that\u0027s all."},{"Start":"14:45.240 ","End":"14:49.585","Text":"Of course, if we know that we\u0027re only dealing with positive x,"},{"Start":"14:49.585 ","End":"14:55.284","Text":"if we\u0027re told, then we can drop the bars and not have absolute."},{"Start":"14:55.284 ","End":"14:58.900","Text":"Other than that, you should use absolute value."},{"Start":"14:58.900 ","End":"15:03.470","Text":"That\u0027s rule number 3 on to the next."}],"ID":1533},{"Watched":false,"Name":"Integration Rules - Part B","Duration":"5m 46s","ChapterTopicVideoID":1520,"CourseChapterTopicPlaylistID":82571,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.com/Images/Videos_Thumbnails/1520.jpeg","UploadDate":"2019-11-14T01:31:43.3470000","DurationForVideoObject":"PT5M46S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:05.475","Text":"Rule number 4 is the integral of e to the power of x dx."},{"Start":"00:05.475 ","End":"00:09.320","Text":"As you remember, the derivative of e to the x is just e to the x."},{"Start":"00:09.320 ","End":"00:13.740","Text":"The integral is also just e to the x plus C."},{"Start":"00:13.740 ","End":"00:18.750","Text":"These 4 are the 4 basic ones we\u0027re going to introduce in this clip."},{"Start":"00:18.750 ","End":"00:20.190","Text":"There are many, many others,"},{"Start":"00:20.190 ","End":"00:23.100","Text":"and most of them can be found in the formula sheets,"},{"Start":"00:23.100 ","End":"00:26.925","Text":"especially not going to go into the trigonometric 1 sine, cosine,"},{"Start":"00:26.925 ","End":"00:30.600","Text":"tangent because not everyone is studying trigonometry but they are"},{"Start":"00:30.600 ","End":"00:34.420","Text":"there in the formula sheets and in the lessons when you learn about them,"},{"Start":"00:34.420 ","End":"00:38.030","Text":"what we do need beyond these basic rules are rules"},{"Start":"00:38.030 ","End":"00:41.735","Text":"of how to combine things or variations of these."},{"Start":"00:41.735 ","End":"00:45.845","Text":"For example, what if I had a 100 e to the x?"},{"Start":"00:45.845 ","End":"00:50.350","Text":"What if I had minus 7x to the 5th?"},{"Start":"00:50.350 ","End":"00:52.320","Text":"What if I had the sum of 2 things?"},{"Start":"00:52.320 ","End":"00:56.870","Text":"What if I had 1 over x plus 3 times e to the x even?"},{"Start":"00:56.870 ","End":"00:59.795","Text":"So these are the kinds of things we need for the Meta rules."},{"Start":"00:59.795 ","End":"01:01.130","Text":"In fact, they\u0027ll be 2 more,"},{"Start":"01:01.130 ","End":"01:04.295","Text":"one of them will deal with a constant times a function,"},{"Start":"01:04.295 ","End":"01:07.970","Text":"and the other will deal with the sum or difference of 2 functions."},{"Start":"01:07.970 ","End":"01:11.720","Text":"Let\u0027s get on to rule number 5 which like I said,"},{"Start":"01:11.720 ","End":"01:13.910","Text":"is a constant times a function."},{"Start":"01:13.910 ","End":"01:20.855","Text":"The rule is that if I have the integral of a constant a times some function of x,"},{"Start":"01:20.855 ","End":"01:27.420","Text":"then this is exactly equal to the constant times the integral of the function."},{"Start":"01:27.420 ","End":"01:32.090","Text":"In practice, what it means is that if there\u0027s a constant stuck to a function,"},{"Start":"01:32.090 ","End":"01:34.220","Text":"I stuck I mean multiplying by,"},{"Start":"01:34.220 ","End":"01:38.345","Text":"we just leave that constant alone and integrate the function."},{"Start":"01:38.345 ","End":"01:40.085","Text":"I\u0027ll give you an example of this."},{"Start":"01:40.085 ","End":"01:45.755","Text":"If I have, let\u0027s say the integral of 4x to the 10th,"},{"Start":"01:45.755 ","End":"01:51.430","Text":"then it\u0027s just 4 times the integral of x to the 10th."},{"Start":"01:51.430 ","End":"01:53.880","Text":"This we know how to do,"},{"Start":"01:53.880 ","End":"02:00.215","Text":"x to the 10th is the integral of it is x to the 11 over 11,"},{"Start":"02:00.215 ","End":"02:02.585","Text":"and the 4 stays here."},{"Start":"02:02.585 ","End":"02:04.730","Text":"Finally, at the end we put a plus C,"},{"Start":"02:04.730 ","End":"02:07.340","Text":"we don\u0027t put the plus C in the intermediate stages,"},{"Start":"02:07.340 ","End":"02:08.915","Text":"just put 1 at the end."},{"Start":"02:08.915 ","End":"02:14.345","Text":"After a while, you learn about the constants and then you do it all in 1 step,"},{"Start":"02:14.345 ","End":"02:18.035","Text":"you see 4x to the 10th and you would straight away,"},{"Start":"02:18.035 ","End":"02:19.755","Text":"skip this step and say,"},{"Start":"02:19.755 ","End":"02:22.535","Text":"I have 4 and I leave the 4 there,"},{"Start":"02:22.535 ","End":"02:24.960","Text":"and then I go x to the 11th over 11th."},{"Start":"02:24.960 ","End":"02:29.645","Text":"You get from here to here all in 1 jump after you\u0027ve got a bit of practice."},{"Start":"02:29.645 ","End":"02:35.615","Text":"Another example is the integral of 10 over x dx."},{"Start":"02:35.615 ","End":"02:38.450","Text":"In this case, I see there\u0027s a constant here,"},{"Start":"02:38.450 ","End":"02:42.155","Text":"10 times 1 over x instead of taking it out and so on,"},{"Start":"02:42.155 ","End":"02:49.340","Text":"I just say the 10 stays there and the integral of 1 over x is natural log of x,"},{"Start":"02:49.340 ","End":"02:51.005","Text":"so the value of x."},{"Start":"02:51.005 ","End":"02:56.945","Text":"At the very end, you add the plus C. That\u0027s rule number 5."},{"Start":"02:56.945 ","End":"03:01.625","Text":"Rule number 6 is going to deal with sums and differences,"},{"Start":"03:01.625 ","End":"03:04.160","Text":"which talks about the sum of 2 functions,"},{"Start":"03:04.160 ","End":"03:10.280","Text":"lets say I have 1 function of x plus another function of x dx."},{"Start":"03:10.280 ","End":"03:13.765","Text":"I just integrate each 1 separately and then add them."},{"Start":"03:13.765 ","End":"03:22.105","Text":"This is the integral of f of x dx plus the integral of g of x dx."},{"Start":"03:22.105 ","End":"03:27.635","Text":"It works also with 3 or more functions if f plus g plus h and so on."},{"Start":"03:27.635 ","End":"03:28.940","Text":"I\u0027ll take for example,"},{"Start":"03:28.940 ","End":"03:32.989","Text":"the integral of x squared plus"},{"Start":"03:32.989 ","End":"03:39.885","Text":"x to the 10th minus x to the 4th dx and I see I have a minus here."},{"Start":"03:39.885 ","End":"03:42.560","Text":"Let me just tell you that same rule works with a minus."},{"Start":"03:42.560 ","End":"03:47.150","Text":"I\u0027ll take the integral of this plus the integral of this minus the integral of this."},{"Start":"03:47.150 ","End":"03:55.475","Text":"In other words, this is equal to the integral of x squared dx plus the integral of"},{"Start":"03:55.475 ","End":"04:00.165","Text":"x to the 10th dx minus the integral of"},{"Start":"04:00.165 ","End":"04:06.510","Text":"x to the 4th dx and this is equal to x cubed over 3."},{"Start":"04:06.510 ","End":"04:15.675","Text":"This is x to the 11th over 11 because there\u0027s a minus x to the 5th over 5."},{"Start":"04:15.675 ","End":"04:19.100","Text":"At the very end we put a plus C. You might be wondering,"},{"Start":"04:19.100 ","End":"04:21.800","Text":"why don\u0027t we put a plus C_1 here,"},{"Start":"04:21.800 ","End":"04:24.185","Text":"C_2 here, C_3 here,"},{"Start":"04:24.185 ","End":"04:26.690","Text":"that\u0027s because all of these constants when you add them together,"},{"Start":"04:26.690 ","End":"04:28.490","Text":"it just gives you a constant at the end,"},{"Start":"04:28.490 ","End":"04:31.280","Text":"there\u0027s no need at every stage to put a plus C,"},{"Start":"04:31.280 ","End":"04:36.080","Text":"we only put 1 single plus C at the end of all these mixed exercises."},{"Start":"04:36.080 ","End":"04:40.810","Text":"Now I\u0027d like to give an example where we combine 5 and 6."},{"Start":"04:40.810 ","End":"04:42.500","Text":"Let\u0027s give an example."},{"Start":"04:42.500 ","End":"04:53.235","Text":"The integral of 4x to the 10th minus a 100x to the 7th dx."},{"Start":"04:53.235 ","End":"04:54.570","Text":"Just 1 more thing,"},{"Start":"04:54.570 ","End":"04:57.680","Text":"when I did this passage from here to here,"},{"Start":"04:57.680 ","End":"05:02.060","Text":"I was just assuming that you\u0027re ready familiar with this rule number 2,"},{"Start":"05:02.060 ","End":"05:04.220","Text":"and this is the rule that I was using all along"},{"Start":"05:04.220 ","End":"05:06.750","Text":"increasing the power by 1 and dividing by it,"},{"Start":"05:06.750 ","End":"05:08.750","Text":"we\u0027ll see that again in here."},{"Start":"05:08.750 ","End":"05:11.555","Text":"Now we\u0027re going to use rules 5 and 6,"},{"Start":"05:11.555 ","End":"05:15.930","Text":"and we usually combine them in our head so we see that we have a constant here,"},{"Start":"05:15.930 ","End":"05:19.055","Text":"we leave the constant there, then we see x to the 10th,"},{"Start":"05:19.055 ","End":"05:21.370","Text":"we use the rule for x to the n,"},{"Start":"05:21.370 ","End":"05:25.350","Text":"which is x to the n plus 1 over n plus 1,"},{"Start":"05:25.350 ","End":"05:26.790","Text":"then we see a minus."},{"Start":"05:26.790 ","End":"05:31.985","Text":"We know that we have to subtract a 100 just stays there from rule 5."},{"Start":"05:31.985 ","End":"05:33.860","Text":"Then using the rule for the x to the n,"},{"Start":"05:33.860 ","End":"05:35.620","Text":"which is lost up there somewhere,"},{"Start":"05:35.620 ","End":"05:37.745","Text":"increase the index by 1,"},{"Start":"05:37.745 ","End":"05:42.240","Text":"x to the 8th over 8 and then plus C. Okay,"},{"Start":"05:42.240 ","End":"05:46.980","Text":"this is all the rules for now and done with this clip."}],"ID":1534},{"Watched":false,"Name":"Integration Rules - Part C","Duration":"4m 19s","ChapterTopicVideoID":1521,"CourseChapterTopicPlaylistID":82571,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.com/Images/Videos_Thumbnails/1521.jpeg","UploadDate":"2017-01-25T23:52:50.5630000","DurationForVideoObject":"PT4M19S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:02.940","Text":"Following the clip on integration rules,"},{"Start":"00:02.940 ","End":"00:05.640","Text":"there are some important words I want to say,"},{"Start":"00:05.640 ","End":"00:09.345","Text":"ending in a warning almost or a strong note."},{"Start":"00:09.345 ","End":"00:11.790","Text":"The rules for integration followed very"},{"Start":"00:11.790 ","End":"00:14.520","Text":"much in the same path as the rules for differentiation."},{"Start":"00:14.520 ","End":"00:16.170","Text":"We had certain basic functions,"},{"Start":"00:16.170 ","End":"00:17.685","Text":"constant x to the n,"},{"Start":"00:17.685 ","End":"00:19.140","Text":"e to the x, and so on,"},{"Start":"00:19.140 ","End":"00:21.210","Text":"and then we had some more general rules."},{"Start":"00:21.210 ","End":"00:23.175","Text":"A constant times a function,"},{"Start":"00:23.175 ","End":"00:26.170","Text":"you leave the constant there and just integrate the function."},{"Start":"00:26.170 ","End":"00:30.080","Text":"This was very similar to differentiation and the same thing with the sum and"},{"Start":"00:30.080 ","End":"00:34.415","Text":"difference of integrals was pretty much the same rule of the differentiation."},{"Start":"00:34.415 ","End":"00:35.990","Text":"The sum becomes a sum,"},{"Start":"00:35.990 ","End":"00:37.310","Text":"the difference becomes a difference."},{"Start":"00:37.310 ","End":"00:39.530","Text":"What we would expect next would be,"},{"Start":"00:39.530 ","End":"00:45.395","Text":"we\u0027d expect rule 7 and 8 to be the product and quotient, multiplication and division."},{"Start":"00:45.395 ","End":"00:51.030","Text":"So you would expect the rule for the integral of f of x times g of x,"},{"Start":"00:51.030 ","End":"00:57.410","Text":"and we would expect to rule for f of x divided by g of x dx."},{"Start":"00:57.410 ","End":"01:03.005","Text":"Then we\u0027d be able to solve problems such as what is the integral"},{"Start":"01:03.005 ","End":"01:09.325","Text":"of x to the fourth times natural log of x,"},{"Start":"01:09.325 ","End":"01:17.105","Text":"and also things like maybe e to the x over natural log of x, and so on."},{"Start":"01:17.105 ","End":"01:19.745","Text":"Here\u0027s the news I have to break to you."},{"Start":"01:19.745 ","End":"01:21.050","Text":"It\u0027s not good news."},{"Start":"01:21.050 ","End":"01:23.674","Text":"There are no such rules."},{"Start":"01:23.674 ","End":"01:26.600","Text":"They do not exist, nor will they."},{"Start":"01:26.600 ","End":"01:30.845","Text":"In fact, I\u0027m going to just completely erase all this."},{"Start":"01:30.845 ","End":"01:34.825","Text":"There are no quotient and product rules in integration."},{"Start":"01:34.825 ","End":"01:37.190","Text":"If you think that\u0027s a good thing because then you\u0027re not going to"},{"Start":"01:37.190 ","End":"01:39.470","Text":"be asked on the exam about and quotients,"},{"Start":"01:39.470 ","End":"01:40.985","Text":"well, you\u0027re on there too."},{"Start":"01:40.985 ","End":"01:43.280","Text":"You will have questions with products and quotients."},{"Start":"01:43.280 ","End":"01:45.320","Text":"We\u0027ll just have to use other techniques."},{"Start":"01:45.320 ","End":"01:50.710","Text":"Mainly we will try and convert some products and quotients to sums and differences."},{"Start":"01:50.710 ","End":"01:53.750","Text":"I\u0027ll write it as a note though it\u0027s more a warning,"},{"Start":"01:53.750 ","End":"02:02.165","Text":"that there are no general rules for the integration of a product or a quotient,"},{"Start":"02:02.165 ","End":"02:04.895","Text":"which means multiplication and division."},{"Start":"02:04.895 ","End":"02:08.900","Text":"Product is multiplication and quotient is division."},{"Start":"02:08.900 ","End":"02:10.930","Text":"You will get products and quotients."},{"Start":"02:10.930 ","End":"02:16.190","Text":"We\u0027ll have to get around them by various tricks and techniques, but nothing general."},{"Start":"02:16.190 ","End":"02:19.775","Text":"As I mentioned, the main idea is to somehow convert"},{"Start":"02:19.775 ","End":"02:23.830","Text":"product or the quotient into sum and difference."},{"Start":"02:23.830 ","End":"02:25.460","Text":"I\u0027ll give an example."},{"Start":"02:25.460 ","End":"02:33.240","Text":"Let\u0027s say I have the integral of x squared plus 1 times x plus 4 dx."},{"Start":"02:33.240 ","End":"02:34.535","Text":"Now, this is a product,"},{"Start":"02:34.535 ","End":"02:36.125","Text":"and how would I handle this?"},{"Start":"02:36.125 ","End":"02:37.880","Text":"Well, that\u0027s not really a problem."},{"Start":"02:37.880 ","End":"02:41.960","Text":"What we can do is open brackets like in algebra."},{"Start":"02:41.960 ","End":"02:43.535","Text":"If we open the brackets,"},{"Start":"02:43.535 ","End":"02:45.155","Text":"we get the integral."},{"Start":"02:45.155 ","End":"02:53.075","Text":"We have x cubed plus 4x squared plus x plus 4 dx."},{"Start":"02:53.075 ","End":"02:54.905","Text":"We have no problem with this."},{"Start":"02:54.905 ","End":"03:01.114","Text":"This is x to the fourth over 4 plus 4x cubed over 3,"},{"Start":"03:01.114 ","End":"03:04.910","Text":"and x is x squared over 2, add 4."},{"Start":"03:04.910 ","End":"03:06.980","Text":"The integral of 1 is just x,"},{"Start":"03:06.980 ","End":"03:11.300","Text":"and finally plus C. So that\u0027s how I might handle this product."},{"Start":"03:11.300 ","End":"03:13.385","Text":"I just convert it into a sum."},{"Start":"03:13.385 ","End":"03:15.740","Text":"How about a quotient?"},{"Start":"03:15.740 ","End":"03:17.425","Text":"Let\u0027s give an example of that."},{"Start":"03:17.425 ","End":"03:25.660","Text":"Let\u0027s have the integral of x squared plus x plus 1 over x dx."},{"Start":"03:25.660 ","End":"03:28.130","Text":"In here, I can also use a trick."},{"Start":"03:28.130 ","End":"03:31.085","Text":"It\u0027s not a trick, it\u0027s just the algebraic distributive law."},{"Start":"03:31.085 ","End":"03:32.690","Text":"When I have the sum of these things,"},{"Start":"03:32.690 ","End":"03:35.000","Text":"divide it all by the same denominator."},{"Start":"03:35.000 ","End":"03:37.280","Text":"I just divide each one by the denominator."},{"Start":"03:37.280 ","End":"03:41.690","Text":"I\u0027m just using algebra here to convert this quotient into a sum."},{"Start":"03:41.690 ","End":"03:44.630","Text":"I get x squared over x is x,"},{"Start":"03:44.630 ","End":"03:47.375","Text":"x over x is 1,"},{"Start":"03:47.375 ","End":"03:50.030","Text":"and 1 over x is just 1 over x."},{"Start":"03:50.030 ","End":"03:52.375","Text":"Now, this I can handle."},{"Start":"03:52.375 ","End":"03:56.165","Text":"The integral of x is x squared over 2,"},{"Start":"03:56.165 ","End":"03:58.310","Text":"integral of 1 is x,"},{"Start":"03:58.310 ","End":"04:01.625","Text":"and integral of 1 over x is the natural log of x,"},{"Start":"04:01.625 ","End":"04:04.040","Text":"or actually the absolute value of x plus"},{"Start":"04:04.040 ","End":"04:08.120","Text":"c. You will be learning a few little tricks that will help,"},{"Start":"04:08.120 ","End":"04:10.745","Text":"especially with the product and some techniques,"},{"Start":"04:10.745 ","End":"04:12.650","Text":"but not a general rule."},{"Start":"04:12.650 ","End":"04:15.245","Text":"I think I\u0027ve said about all I want to say about"},{"Start":"04:15.245 ","End":"04:19.680","Text":"products and quotients and integration. That\u0027s it."}],"ID":1535},{"Watched":false,"Name":"Exercise 1","Duration":"1m 18s","ChapterTopicVideoID":1502,"CourseChapterTopicPlaylistID":82571,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.com/Images/Videos_Thumbnails/1502.jpeg","UploadDate":"2016-06-26T05:35:00.6300000","DurationForVideoObject":"PT1M18S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:04.320","Text":"In this exercise, we have to compute the following integrals,"},{"Start":"00:04.320 ","End":"00:05.475","Text":"and there are 3 of them."},{"Start":"00:05.475 ","End":"00:06.660","Text":"In all 3 of these,"},{"Start":"00:06.660 ","End":"00:08.385","Text":"we\u0027d need the same formula,"},{"Start":"00:08.385 ","End":"00:13.590","Text":"and that formula is that the integral of a dx"},{"Start":"00:13.590 ","End":"00:19.935","Text":"is equal to ax plus c. In the first 1,"},{"Start":"00:19.935 ","End":"00:21.680","Text":"the a is 4,"},{"Start":"00:21.680 ","End":"00:29.340","Text":"and so the integral of 4dx is equal to just 4x plus c. There\u0027s always the constant there,"},{"Start":"00:29.340 ","End":"00:32.550","Text":"the plus C, all the indefinite integrals have this."},{"Start":"00:32.550 ","End":"00:34.245","Text":"The next 1, B."},{"Start":"00:34.245 ","End":"00:38.985","Text":"The a here, is all this 4e over square root of 5,"},{"Start":"00:38.985 ","End":"00:43.250","Text":"so the integral of this 4e over the square root of 5,"},{"Start":"00:43.250 ","End":"00:45.005","Text":"which is just a constant,"},{"Start":"00:45.005 ","End":"00:46.625","Text":"is just the same thing,"},{"Start":"00:46.625 ","End":"00:49.744","Text":"4e over the square root of 5,"},{"Start":"00:49.744 ","End":"00:52.730","Text":"but times x plus c,"},{"Start":"00:52.730 ","End":"00:54.760","Text":"just using the same formula."},{"Start":"00:54.760 ","End":"00:58.250","Text":"In question C, we don\u0027t see a constant a,"},{"Start":"00:58.250 ","End":"00:59.770","Text":"so we just put a 1 there,"},{"Start":"00:59.770 ","End":"01:07.115","Text":"so what we have is that the integral of just dx is actually the integral of 1 dx."},{"Start":"01:07.115 ","End":"01:09.665","Text":"Now, 1 is our a here,"},{"Start":"01:09.665 ","End":"01:11.960","Text":"so it\u0027s 1x plus c,"},{"Start":"01:11.960 ","End":"01:13.900","Text":"but we don\u0027t just write 1x,"},{"Start":"01:13.900 ","End":"01:17.405","Text":"we just write x, so it\u0027s x plus c,"},{"Start":"01:17.405 ","End":"01:19.710","Text":"and that\u0027s all there is to it."}],"ID":1492},{"Watched":false,"Name":"Exercise 2","Duration":"2m 38s","ChapterTopicVideoID":1503,"CourseChapterTopicPlaylistID":82571,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.com/Images/Videos_Thumbnails/1503.jpeg","UploadDate":"2016-06-26T05:35:16.9270000","DurationForVideoObject":"PT2M38S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:01.010 ","End":"00:05.505","Text":"In this exercise, we have to compute the following integrals."},{"Start":"00:05.505 ","End":"00:08.910","Text":"Let\u0027s start with the first one."},{"Start":"00:08.910 ","End":"00:13.455","Text":"The integral of x to the minus 3 dx."},{"Start":"00:13.455 ","End":"00:15.675","Text":"We\u0027ll need the standard formula,"},{"Start":"00:15.675 ","End":"00:17.790","Text":"the basic integration formula,"},{"Start":"00:17.790 ","End":"00:22.710","Text":"that the integral of x to the n dx is equal"},{"Start":"00:22.710 ","End":"00:29.085","Text":"to x to the n plus 1 over n plus 1 and as always,"},{"Start":"00:29.085 ","End":"00:31.530","Text":"plus C. In this case,"},{"Start":"00:31.530 ","End":"00:35.085","Text":"we have n equals negative 3."},{"Start":"00:35.085 ","End":"00:45.410","Text":"We have x to the power of minus 3 plus 1 over minus 3 plus 1 plus C,"},{"Start":"00:45.410 ","End":"00:49.790","Text":"which equals x to the power of minus"},{"Start":"00:49.790 ","End":"00:56.135","Text":"2 over minus 2 plus C. That\u0027s the answer."},{"Start":"00:56.135 ","End":"01:01.670","Text":"Next one, integral of 1 over x to the 4 dx."},{"Start":"01:01.670 ","End":"01:05.210","Text":"In this case, we\u0027re also going to use this formula,"},{"Start":"01:05.210 ","End":"01:08.545","Text":"but we have to do a tiny bit of algebra first."},{"Start":"01:08.545 ","End":"01:14.975","Text":"Remember, that 1 over x to anything, let\u0027s say n,"},{"Start":"01:14.975 ","End":"01:19.760","Text":"is equal to x to the minus n. Here we\u0027ll change it"},{"Start":"01:19.760 ","End":"01:24.825","Text":"to the integral of x to the minus 4 dx."},{"Start":"01:24.825 ","End":"01:27.875","Text":"Now, it does look like this rule up here,"},{"Start":"01:27.875 ","End":"01:30.560","Text":"but with n equals minus 4."},{"Start":"01:30.560 ","End":"01:39.020","Text":"After the integration, we have x to the minus 4 plus 1 over minus 4 plus 1,"},{"Start":"01:39.020 ","End":"01:40.850","Text":"and as always, plus C,"},{"Start":"01:40.850 ","End":"01:45.935","Text":"which equals x to the minus 3 over"},{"Start":"01:45.935 ","End":"01:51.440","Text":"minus 3 plus C. The last 1 is like this,"},{"Start":"01:51.440 ","End":"01:54.965","Text":"but it\u0027s just 1 over x to the 10."},{"Start":"01:54.965 ","End":"02:00.265","Text":"The integral of 1 over x to the 10 dx,"},{"Start":"02:00.265 ","End":"02:01.910","Text":"almost exactly like this,"},{"Start":"02:01.910 ","End":"02:03.890","Text":"except with 10 replacing the 4."},{"Start":"02:03.890 ","End":"02:06.215","Text":"We first of all rewrite it in the form,"},{"Start":"02:06.215 ","End":"02:13.640","Text":"integral of x to the minus 10 dx by the same algebraic rule here."},{"Start":"02:13.640 ","End":"02:16.160","Text":"Then using the integration formula,"},{"Start":"02:16.160 ","End":"02:18.380","Text":"we get that this is equal to,"},{"Start":"02:18.380 ","End":"02:20.420","Text":"where n equals minus 10,"},{"Start":"02:20.420 ","End":"02:29.415","Text":"x to the power of minus 10 plus 1 over minus 10 plus 1 and plus C, let\u0027s not forget."},{"Start":"02:29.415 ","End":"02:39.040","Text":"Finally, x to the minus 9 over minus 9 plus C. That\u0027s the answer to the third 1."}],"ID":1493},{"Watched":false,"Name":"Exercise 3","Duration":"3m 55s","ChapterTopicVideoID":1504,"CourseChapterTopicPlaylistID":82571,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.com/Images/Videos_Thumbnails/1504.jpeg","UploadDate":"2016-06-26T05:35:41.0300000","DurationForVideoObject":"PT3M55S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:04.155","Text":"In this exercise, we have to compute the following integrals."},{"Start":"00:04.155 ","End":"00:09.105","Text":"Looking ahead at them, I see that there\u0027s 1 formula I can use in all 3."},{"Start":"00:09.105 ","End":"00:17.190","Text":"That is that the integral of x to the power of n dx is equal to,"},{"Start":"00:17.190 ","End":"00:19.110","Text":"provided then is not minus 1,"},{"Start":"00:19.110 ","End":"00:21.345","Text":"which it isn\u0027t in any of these cases."},{"Start":"00:21.345 ","End":"00:26.415","Text":"It\u0027s equal to x to the n plus 1 over n plus 1."},{"Start":"00:26.415 ","End":"00:29.385","Text":"There\u0027s always a plus C at the end."},{"Start":"00:29.385 ","End":"00:32.590","Text":"Let\u0027s get to the first 1."},{"Start":"00:32.590 ","End":"00:37.790","Text":"Integral of x to the power of a 1/4 dx is equal to."},{"Start":"00:37.790 ","End":"00:39.454","Text":"Just using this formula,"},{"Start":"00:39.454 ","End":"00:41.815","Text":"replacing n by 1/4,"},{"Start":"00:41.815 ","End":"00:47.544","Text":"we get x to the power of 1/4 plus 1"},{"Start":"00:47.544 ","End":"00:54.965","Text":"over 1/4 plus 1 plus C. I can rewrite this,"},{"Start":"00:54.965 ","End":"00:56.195","Text":"this is just fine."},{"Start":"00:56.195 ","End":"01:00.275","Text":"I could rewrite it as x to the power of 5/4"},{"Start":"01:00.275 ","End":"01:06.650","Text":"over 5/4 plus C. Of course you could do more algebra,"},{"Start":"01:06.650 ","End":"01:08.570","Text":"and write this as 4/5 and so on,"},{"Start":"01:08.570 ","End":"01:10.415","Text":"but this is adequate."},{"Start":"01:10.415 ","End":"01:17.370","Text":"The next integral we have to solve is the square root of x dx."},{"Start":"01:17.370 ","End":"01:19.895","Text":"Before we use this formula,"},{"Start":"01:19.895 ","End":"01:23.960","Text":"we have to rewrite this and remember that the square root of"},{"Start":"01:23.960 ","End":"01:29.195","Text":"x from algebra is just x to the power of 1.5."},{"Start":"01:29.195 ","End":"01:31.625","Text":"If this is x to the power of 1.5,"},{"Start":"01:31.625 ","End":"01:36.170","Text":"we can now use this with n equals 1.5."},{"Start":"01:36.170 ","End":"01:41.210","Text":"We can say that this is equal to x to the power of"},{"Start":"01:41.210 ","End":"01:47.110","Text":"1.5 plus 1 over 1.5 plus 1."},{"Start":"01:47.110 ","End":"01:56.760","Text":"Then plus C which is equal to x to the power of 3 over 2 over 3 over"},{"Start":"01:56.760 ","End":"02:00.940","Text":"2 plus C. Since"},{"Start":"02:00.940 ","End":"02:03.500","Text":"the original question was given in terms of"},{"Start":"02:03.500 ","End":"02:06.505","Text":"the square root and not a fractional exponent,"},{"Start":"02:06.505 ","End":"02:10.640","Text":"we can be nice and put it in that form also."},{"Start":"02:10.640 ","End":"02:12.890","Text":"I\u0027ll just give you the answer."},{"Start":"02:12.890 ","End":"02:15.395","Text":"I\u0027ll let you figure out the algebra."},{"Start":"02:15.395 ","End":"02:19.750","Text":"This is equal to x times square root of x."},{"Start":"02:19.750 ","End":"02:22.730","Text":"The fraction I could leave in the denominator,"},{"Start":"02:22.730 ","End":"02:27.710","Text":"but I\u0027d like to invert it and put it on the numerator like 2/3 again plus"},{"Start":"02:27.710 ","End":"02:35.120","Text":"C. The last question was the integral of the 4th root of x cubed."},{"Start":"02:35.120 ","End":"02:41.735","Text":"We have the integral of the 4th root of x cubed, dx."},{"Start":"02:41.735 ","End":"02:46.580","Text":"We need that, bring it algebraically to the form x to the n. Again,"},{"Start":"02:46.580 ","End":"02:48.260","Text":"if you remember your algebra,"},{"Start":"02:48.260 ","End":"02:51.245","Text":"when you have these kinds of expressions,"},{"Start":"02:51.245 ","End":"02:53.300","Text":"square roots of powers,"},{"Start":"02:53.300 ","End":"02:56.405","Text":"then this is just x to the power of 3 over 4."},{"Start":"02:56.405 ","End":"02:58.010","Text":"This puts it in the numerator,"},{"Start":"02:58.010 ","End":"02:59.765","Text":"this puts it in the denominator."},{"Start":"02:59.765 ","End":"03:04.375","Text":"This is x to the power of 3 over 4dx."},{"Start":"03:04.375 ","End":"03:12.035","Text":"Now I can go ahead and use this formula with n equals 3/4 by that formula,"},{"Start":"03:12.035 ","End":"03:16.400","Text":"x to the power of 3/4 plus 1"},{"Start":"03:16.400 ","End":"03:22.970","Text":"over 3/4 plus 1 and then plus C. But finally,"},{"Start":"03:22.970 ","End":"03:26.210","Text":"I would want to bring it back to this form."},{"Start":"03:26.210 ","End":"03:27.995","Text":"I\u0027ll write that at the side;"},{"Start":"03:27.995 ","End":"03:31.415","Text":"3/4 plus 1 is 7/4."},{"Start":"03:31.415 ","End":"03:33.050","Text":"Now I can write it."},{"Start":"03:33.050 ","End":"03:34.850","Text":"If this is 7/4,"},{"Start":"03:34.850 ","End":"03:42.125","Text":"I can put it in the front as 4/7 and then times x to the power of 7 over 4."},{"Start":"03:42.125 ","End":"03:43.760","Text":"With the similar principle,"},{"Start":"03:43.760 ","End":"03:47.365","Text":"it\u0027ll be the 4th root of x to the 7th."},{"Start":"03:47.365 ","End":"03:49.635","Text":"That would be x to the 7 over 4."},{"Start":"03:49.635 ","End":"03:51.465","Text":"Let\u0027s not forget finally,"},{"Start":"03:51.465 ","End":"03:56.290","Text":"the plus C. That\u0027s all 3 of them done."}],"ID":1494},{"Watched":false,"Name":"Exercise 4","Duration":"4m 49s","ChapterTopicVideoID":1505,"CourseChapterTopicPlaylistID":82571,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.com/Images/Videos_Thumbnails/1505.jpeg","UploadDate":"2016-06-26T05:36:09.8130000","DurationForVideoObject":"PT4M49S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:02.710","Text":"In this exercise, we have to compute"},{"Start":"00:02.710 ","End":"00:05.110","Text":"the following integrals."},{"Start":"00:05.110 ","End":"00:06.070","Text":"In all of these,"},{"Start":"00:06.070 ","End":"00:08.170","Text":"we\u0027ll be using the same formula,"},{"Start":"00:08.170 ","End":"00:13.540","Text":"that is that the integral of x^n dx"},{"Start":"00:13.540 ","End":"00:16.675","Text":"is equal to x^n plus 1"},{"Start":"00:16.675 ","End":"00:19.950","Text":"over n plus 1 plus C."},{"Start":"00:19.950 ","End":"00:22.000","Text":"Of course, this only works"},{"Start":"00:22.000 ","End":"00:24.805","Text":"when n is not equal to minus 1,"},{"Start":"00:24.805 ","End":"00:27.625","Text":"which it won\u0027t be in any of these 3 cases,"},{"Start":"00:27.625 ","End":"00:29.785","Text":"n is not equal to minus 1."},{"Start":"00:29.785 ","End":"00:31.810","Text":"What we have to do is some algebra"},{"Start":"00:31.810 ","End":"00:35.350","Text":"to get each of these to look like x^n."},{"Start":"00:35.350 ","End":"00:38.590","Text":"In the first 1, we have the integral"},{"Start":"00:38.590 ","End":"00:42.750","Text":"of 1 over x square root of x dx."},{"Start":"00:42.750 ","End":"00:45.335","Text":"We want to bring this to the form x^n,"},{"Start":"00:45.335 ","End":"00:48.470","Text":"so this is equal 1 over x,"},{"Start":"00:48.470 ","End":"00:52.200","Text":"and the square root of x is x^1/5,"},{"Start":"00:52.300 ","End":"00:55.670","Text":"and x is x^1."},{"Start":"00:55.670 ","End":"00:58.729","Text":"This is equal to the integral"},{"Start":"00:58.729 ","End":"01:04.990","Text":"of 1/x^1 plus a half which is 1/2 dx."},{"Start":"01:04.990 ","End":"01:10.100","Text":"Using the rule that 1/a^n"},{"Start":"01:10.100 ","End":"01:12.880","Text":"is a to the minus n,"},{"Start":"01:12.880 ","End":"01:14.570","Text":"we can get that this is equal"},{"Start":"01:14.570 ","End":"01:16.100","Text":"to the integral of x"},{"Start":"01:16.100 ","End":"01:19.955","Text":"to the power of minus 1/2 dx."},{"Start":"01:19.955 ","End":"01:23.090","Text":"Now, we\u0027re ready to use this rule,"},{"Start":"01:23.090 ","End":"01:24.985","Text":"because it is in this form,"},{"Start":"01:24.985 ","End":"01:28.130","Text":"and so this becomes x to the power"},{"Start":"01:28.130 ","End":"01:33.300","Text":"of minus 1/2 plus 1 is minus a half,"},{"Start":"01:33.300 ","End":"01:34.850","Text":"and this same exponent"},{"Start":"01:34.850 ","End":"01:37.625","Text":"also goes in the denominator,"},{"Start":"01:37.625 ","End":"01:41.270","Text":"and that\u0027s minus 1/2 plus C."},{"Start":"01:41.270 ","End":"01:45.620","Text":"Finally, this is the answer,"},{"Start":"01:45.620 ","End":"01:48.530","Text":"but it\u0027s nice to do a bit of simplification."},{"Start":"01:48.530 ","End":"01:52.775","Text":"So 1 over minus a half is minus 2,"},{"Start":"01:52.775 ","End":"01:58.700","Text":"and x to the minus a half is 1/x^ a half,"},{"Start":"01:58.700 ","End":"02:01.865","Text":"and x^ a half is just the square root of x."},{"Start":"02:01.865 ","End":"02:03.620","Text":"Well, if we like, we can also write it"},{"Start":"02:03.620 ","End":"02:07.925","Text":"as minus 2 over the square root of x,"},{"Start":"02:07.925 ","End":"02:10.055","Text":"and this is the answer."},{"Start":"02:10.055 ","End":"02:11.810","Text":"Now, the next 1,"},{"Start":"02:11.810 ","End":"02:14.570","Text":"the integral of 1 over"},{"Start":"02:14.570 ","End":"02:18.185","Text":"the cube root of x squared dx."},{"Start":"02:18.185 ","End":"02:20.120","Text":"Again, we want to use this formula,"},{"Start":"02:20.120 ","End":"02:22.355","Text":"so we\u0027re going to have to do a bit of algebra."},{"Start":"02:22.355 ","End":"02:23.600","Text":"You\u0027re getting good at this,"},{"Start":"02:23.600 ","End":"02:26.240","Text":"so I\u0027ll just take some shortcuts."},{"Start":"02:26.240 ","End":"02:29.015","Text":"This is x^2/3."},{"Start":"02:29.015 ","End":"02:30.050","Text":"When you have the cube root"},{"Start":"02:30.050 ","End":"02:31.620","Text":"of x squared, it\u0027s x to the 2/3."},{"Start":"02:31.620 ","End":"02:33.985","Text":"But it\u0027s in the denominator,"},{"Start":"02:33.985 ","End":"02:36.500","Text":"so we can write x to the power"},{"Start":"02:36.500 ","End":"02:39.950","Text":"of minus 2 over 3 dx."},{"Start":"02:39.950 ","End":"02:41.990","Text":"Now, that we\u0027ve done that algebra,"},{"Start":"02:41.990 ","End":"02:44.150","Text":"we can now use this formula."},{"Start":"02:44.150 ","End":"02:45.860","Text":"This will be equal to"},{"Start":"02:45.860 ","End":"02:49.265","Text":"minus 2/3 plus 1 is 1/3,"},{"Start":"02:49.265 ","End":"02:53.720","Text":"so it\u0027s x^1/3 over 1/3,"},{"Start":"02:53.720 ","End":"02:56.580","Text":"and plus C of course."},{"Start":"02:56.690 ","End":"02:58.700","Text":"This is okay."},{"Start":"02:58.700 ","End":"03:00.320","Text":"But since the original was given"},{"Start":"03:00.320 ","End":"03:02.525","Text":"in terms of cube roots and so forth,"},{"Start":"03:02.525 ","End":"03:04.940","Text":"we\u0027ll write this in that form also."},{"Start":"03:04.940 ","End":"03:06.560","Text":"This is equal to x to the third"},{"Start":"03:06.560 ","End":"03:08.285","Text":"is the cube root of x."},{"Start":"03:08.285 ","End":"03:09.815","Text":"When you divide by a third,"},{"Start":"03:09.815 ","End":"03:11.690","Text":"the 3 comes to the numerator,"},{"Start":"03:11.690 ","End":"03:15.765","Text":"so it\u0027s 3 times the cube root of x,"},{"Start":"03:15.765 ","End":"03:17.225","Text":"again plus C,"},{"Start":"03:17.225 ","End":"03:19.580","Text":"and that\u0027s the answer to b."},{"Start":"03:19.580 ","End":"03:21.560","Text":"Then finally, we have"},{"Start":"03:21.560 ","End":"03:24.200","Text":"x squared over square root of x,"},{"Start":"03:24.200 ","End":"03:26.225","Text":"the integral of that, dx."},{"Start":"03:26.225 ","End":"03:27.890","Text":"Do some algebra first,"},{"Start":"03:27.890 ","End":"03:33.050","Text":"this is the integral of x squared over x^1/2,"},{"Start":"03:33.050 ","End":"03:35.925","Text":"which is the square root dx."},{"Start":"03:35.925 ","End":"03:38.820","Text":"Now, from algebra, when you divide"},{"Start":"03:38.820 ","End":"03:40.980","Text":"exponents and you subtract them,"},{"Start":"03:40.980 ","End":"03:44.385","Text":"here we have x^2 divided by x^ a half,"},{"Start":"03:44.385 ","End":"03:47.330","Text":"so it\u0027s x^2 minus a half,"},{"Start":"03:47.330 ","End":"03:50.550","Text":"and 2 minus a a half is 1/2."},{"Start":"03:50.550 ","End":"03:53.355","Text":"We have the integral of that, dx."},{"Start":"03:53.355 ","End":"03:56.750","Text":"Now, we can use this formula for x^n."},{"Start":"03:56.750 ","End":"03:59.300","Text":"We raise the exponent by 1,"},{"Start":"03:59.300 ","End":"04:01.675","Text":"so it becomes x^2.5,"},{"Start":"04:01.675 ","End":"04:05.330","Text":"divided by 2.5 plus C."},{"Start":"04:05.330 ","End":"04:09.375","Text":"It would be nice to simplify this a bit."},{"Start":"04:09.375 ","End":"04:12.510","Text":"Well, let\u0027s do it aside here 1/2.5,"},{"Start":"04:12.510 ","End":"04:21.770","Text":"2.5 is 5/2, is 1/5 over 2, which is 2/5."},{"Start":"04:21.770 ","End":"04:29.735","Text":"It\u0027s 2/5, and now x^2.5 is x^2 times x to the power of 0.5,"},{"Start":"04:29.735 ","End":"04:32.375","Text":"because 2.5 is 2 plus 0.5,"},{"Start":"04:32.375 ","End":"04:34.825","Text":"and x to the 0.5 is square root of x."},{"Start":"04:34.825 ","End":"04:38.085","Text":"So it\u0027s x squared times square root of x,"},{"Start":"04:38.085 ","End":"04:41.475","Text":"so it\u0027s 2/5 times x squared,"},{"Start":"04:41.475 ","End":"04:43.095","Text":"square root of x,"},{"Start":"04:43.095 ","End":"04:44.940","Text":"and again, plus C."},{"Start":"04:44.940 ","End":"04:48.380","Text":"This is the answer to the third part,"},{"Start":"04:48.380 ","End":"04:50.670","Text":"and we\u0027re done."}],"ID":1495},{"Watched":false,"Name":"Exercise 5","Duration":"7m 8s","ChapterTopicVideoID":1506,"CourseChapterTopicPlaylistID":82571,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.com/Images/Videos_Thumbnails/1506.jpeg","UploadDate":"2016-06-26T05:36:54.7930000","DurationForVideoObject":"PT7M8S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:04.320","Text":"In this exercise, we have to compute the following integrals."},{"Start":"00:04.320 ","End":"00:10.275","Text":"As before, we\u0027re going to be using the rule which should be already memorized completely,"},{"Start":"00:10.275 ","End":"00:20.445","Text":"that the integral of x^n dx is equal to x^n plus 1 over n plus 1 plus C,"},{"Start":"00:20.445 ","End":"00:25.210","Text":"provided that n is not equal to minus 1."},{"Start":"00:27.860 ","End":"00:31.890","Text":"I\u0027m not going to go through all the stages because we\u0027ve done plenty of these,"},{"Start":"00:31.890 ","End":"00:36.590","Text":"but there\u0027s also another couple of rules about integrals is that,"},{"Start":"00:36.590 ","End":"00:39.370","Text":"if you have a constant times a function,"},{"Start":"00:39.370 ","End":"00:41.810","Text":"and I\u0027m not going to write the whole formula here,"},{"Start":"00:41.810 ","End":"00:45.740","Text":"you just leave the constant there and you integrate the function."},{"Start":"00:45.740 ","End":"00:48.010","Text":"There\u0027s also a rule,"},{"Start":"00:48.010 ","End":"00:50.430","Text":"a meta-rule if you like about functions that,"},{"Start":"00:50.430 ","End":"00:56.075","Text":"when you have the sum of 2 and you integrate each separately and that works fine."},{"Start":"00:56.075 ","End":"00:58.145","Text":"Similarly, if it\u0027s a minus,"},{"Start":"00:58.145 ","End":"01:01.684","Text":"the minus just stays and you integrate each piece separately."},{"Start":"01:01.684 ","End":"01:06.065","Text":"With this rule and with these 2 laws of integration, if you like,"},{"Start":"01:06.065 ","End":"01:08.315","Text":"we\u0027ll go ahead and start with the first,"},{"Start":"01:08.315 ","End":"01:15.350","Text":"which is the integral of 2x squared minus x plus 1 dx."},{"Start":"01:15.350 ","End":"01:18.005","Text":"This is equal to,"},{"Start":"01:18.005 ","End":"01:21.995","Text":"now we have the sum and difference of 3 separate terms,"},{"Start":"01:21.995 ","End":"01:23.855","Text":"so we\u0027ll do each 1 separately."},{"Start":"01:23.855 ","End":"01:25.700","Text":"Now for the 2x squared,"},{"Start":"01:25.700 ","End":"01:29.585","Text":"the 2 just stays and we have to do the integral of x squared."},{"Start":"01:29.585 ","End":"01:35.205","Text":"Integral of x squared from this law is we raise the power by 1,"},{"Start":"01:35.205 ","End":"01:39.420","Text":"so it becomes x^3 and divide by that 3,"},{"Start":"01:39.420 ","End":"01:40.695","Text":"so that\u0027s for the first bit."},{"Start":"01:40.695 ","End":"01:42.239","Text":"Now we see a minus,"},{"Start":"01:42.239 ","End":"01:44.070","Text":"what I said about plus and minus,"},{"Start":"01:44.070 ","End":"01:50.790","Text":"the minus just stays and now we have x. X is x^1, of course."},{"Start":"01:50.790 ","End":"01:53.230","Text":"We just raise the power by 1,"},{"Start":"01:53.230 ","End":"01:58.175","Text":"so it\u0027s x squared and divide by that, we choose over 2."},{"Start":"01:58.175 ","End":"01:59.855","Text":"For the last 1,"},{"Start":"01:59.855 ","End":"02:04.160","Text":"the integral of a constant is just that constant times x."},{"Start":"02:04.160 ","End":"02:08.570","Text":"That\u0027s another law that the integral of a,"},{"Start":"02:08.570 ","End":"02:09.995","Text":"which is a constant,"},{"Start":"02:09.995 ","End":"02:18.965","Text":"is just ax plus C. This actually works if you use this rule with n equals 0, but anyway."},{"Start":"02:18.965 ","End":"02:22.760","Text":"This is plus 1 times x, so it\u0027s 1x,"},{"Start":"02:22.760 ","End":"02:25.130","Text":"I\u0027m just writing it as x and as always,"},{"Start":"02:25.130 ","End":"02:27.505","Text":"we put a C at the end."},{"Start":"02:27.505 ","End":"02:30.775","Text":"That\u0027s the answer in just 1 go."},{"Start":"02:30.775 ","End":"02:39.955","Text":"Now, the next 1 is the integral of 3 over x^4 plus twice the cube root of xdx."},{"Start":"02:39.955 ","End":"02:43.840","Text":"Now here we should do a bit of algebra first"},{"Start":"02:43.840 ","End":"02:48.055","Text":"because we want to get it into the form we can use this rule."},{"Start":"02:48.055 ","End":"02:50.080","Text":"This is the integral."},{"Start":"02:50.080 ","End":"02:55.375","Text":"Now this first 1 we\u0027re using the law of exponents,"},{"Start":"02:55.375 ","End":"03:01.810","Text":"that would be something like 1 over a^n is equal to a^minus n. Again,"},{"Start":"03:01.810 ","End":"03:02.980","Text":"we\u0027ve used this so many times,"},{"Start":"03:02.980 ","End":"03:04.450","Text":"I shouldn\u0027t have to write it."},{"Start":"03:04.450 ","End":"03:08.920","Text":"But this is 3 times x^minus 4"},{"Start":"03:08.920 ","End":"03:15.800","Text":"plus the cube root of x is just x^1/3 dx."},{"Start":"03:15.800 ","End":"03:19.765","Text":"Now that we\u0027ve brought it into the form of x^n,"},{"Start":"03:19.765 ","End":"03:22.960","Text":"each piece, we can use the rules here."},{"Start":"03:22.960 ","End":"03:27.260","Text":"X^minus 4 is, we raise the power by 1,"},{"Start":"03:27.260 ","End":"03:28.830","Text":"we leave the 3 as is."},{"Start":"03:28.830 ","End":"03:30.510","Text":"Raise the power by 1,"},{"Start":"03:30.510 ","End":"03:38.210","Text":"that\u0027s x^minus 3 and divide by that same minus 3 and then there\u0027s a plus here,"},{"Start":"03:38.210 ","End":"03:39.875","Text":"so we leave it as a plus,"},{"Start":"03:39.875 ","End":"03:41.345","Text":"and we do the next bit."},{"Start":"03:41.345 ","End":"03:43.430","Text":"Here, we raise the power by 1,"},{"Start":"03:43.430 ","End":"03:50.450","Text":"so it\u0027s x^1 and 1/3 or 4/3 and divide by that same power,"},{"Start":"03:50.450 ","End":"03:55.700","Text":"4/3 plus the C. Then the optional simplification,"},{"Start":"03:55.700 ","End":"03:57.095","Text":"but I recommend doing it."},{"Start":"03:57.095 ","End":"04:03.600","Text":"3 over minus 3 is minus 1 and x^minus 3 is 1 over x^3."},{"Start":"04:03.600 ","End":"04:09.020","Text":"Here we have minus 1 over x^3 and here we have plus."},{"Start":"04:09.020 ","End":"04:13.580","Text":"Now the fraction which is on the denominator is just inverted,"},{"Start":"04:13.580 ","End":"04:21.515","Text":"so it becomes 3/4 and x^4/3 is x^1 and 1/3,"},{"Start":"04:21.515 ","End":"04:26.395","Text":"which is x^1, times x^1/3,"},{"Start":"04:26.395 ","End":"04:32.940","Text":"and this is just equal to x times the cube root of x."},{"Start":"04:32.940 ","End":"04:38.850","Text":"Coming back here, we have 3/4x times cube"},{"Start":"04:38.850 ","End":"04:44.895","Text":"root of x plus the C. That\u0027s the answer to the second 1."},{"Start":"04:44.895 ","End":"04:47.810","Text":"For the third 1, part c,"},{"Start":"04:47.810 ","End":"04:55.660","Text":"is the integral of x squared plus 1 in brackets, all squared dx."},{"Start":"04:55.660 ","End":"05:02.135","Text":"This is a bit of a tricky 1 because you\u0027re liable to think that if the"},{"Start":"05:02.135 ","End":"05:09.735","Text":"integral of x squared dx is x^3 over 3 plus C,"},{"Start":"05:09.735 ","End":"05:15.170","Text":"you might think that the integral of something squared would be that"},{"Start":"05:15.170 ","End":"05:20.960","Text":"something cubed over 3 plus C and you\u0027d be wrong."},{"Start":"05:20.960 ","End":"05:23.810","Text":"There is no template rule for something squared,"},{"Start":"05:23.810 ","End":"05:26.645","Text":"so you can\u0027t do that, so delete that."},{"Start":"05:26.645 ","End":"05:29.810","Text":"Instead you have to use some other technique."},{"Start":"05:29.810 ","End":"05:33.040","Text":"What we\u0027re going to do is just use a bit of algebra."},{"Start":"05:33.040 ","End":"05:34.460","Text":"So instead of this,"},{"Start":"05:34.460 ","End":"05:38.960","Text":"we\u0027re just going to use algebra and we\u0027re going to use our old friend shortcut."},{"Start":"05:38.960 ","End":"05:46.925","Text":"It\u0027s a plus b squared is equal to a squared plus 2ab plus b squared."},{"Start":"05:46.925 ","End":"05:48.590","Text":"Since we have a sum squared,"},{"Start":"05:48.590 ","End":"05:50.930","Text":"we\u0027ll do that with algebra and then you\u0027ll be able to"},{"Start":"05:50.930 ","End":"05:53.825","Text":"use this rule which are always using."},{"Start":"05:53.825 ","End":"05:56.030","Text":"This is equal to the integral."},{"Start":"05:56.030 ","End":"05:58.984","Text":"Now, a squared is x squared squared,"},{"Start":"05:58.984 ","End":"06:01.790","Text":"which is just x^4,"},{"Start":"06:01.790 ","End":"06:06.345","Text":"plus 2ab twice x squared times 1,"},{"Start":"06:06.345 ","End":"06:10.290","Text":"which is just 2x squared plus b squared,"},{"Start":"06:10.290 ","End":"06:14.790","Text":"which is 1 squared and 1 squared is 1 dx."},{"Start":"06:14.790 ","End":"06:19.100","Text":"Now each of the terms here is 1 of these,"},{"Start":"06:19.100 ","End":"06:23.180","Text":"either with a sum or a difference and with a constant."},{"Start":"06:23.180 ","End":"06:28.560","Text":"What we\u0027ll do is say that this is equal to, now, here\u0027s x^4,"},{"Start":"06:28.560 ","End":"06:29.865","Text":"so using this law,"},{"Start":"06:29.865 ","End":"06:36.135","Text":"we just get x^5 over 5 by using this."},{"Start":"06:36.135 ","End":"06:38.760","Text":"Then we have 2x squared,"},{"Start":"06:38.760 ","End":"06:40.140","Text":"so where there\u0027s a constant,"},{"Start":"06:40.140 ","End":"06:45.585","Text":"the constant just stays and we apply this rule to the x squared,"},{"Start":"06:45.585 ","End":"06:50.520","Text":"so we get x^3 over 3,"},{"Start":"06:50.520 ","End":"06:52.260","Text":"and then we have a constant,"},{"Start":"06:52.260 ","End":"06:54.515","Text":"so we use the rule for the constant."},{"Start":"06:54.515 ","End":"06:57.680","Text":"The integral of a constant is that constant times x."},{"Start":"06:57.680 ","End":"06:59.600","Text":"So it\u0027s 1 times x,"},{"Start":"06:59.600 ","End":"07:01.380","Text":"which is just x,"},{"Start":"07:01.380 ","End":"07:06.425","Text":"and finally, plus the constant and that\u0027s the answer."},{"Start":"07:06.425 ","End":"07:09.690","Text":"We\u0027re done with all 3 and that\u0027s it."}],"ID":1496},{"Watched":false,"Name":"Exercise 6","Duration":"6m 33s","ChapterTopicVideoID":6638,"CourseChapterTopicPlaylistID":82571,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.com/Images/Videos_Thumbnails/6638.jpeg","UploadDate":"2016-07-19T11:46:51.1730000","DurationForVideoObject":"PT6M33S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:04.650","Text":"In this exercise, we have to compute the following integrals."},{"Start":"00:04.650 ","End":"00:07.760","Text":"The first 1 just copied it from here,"},{"Start":"00:07.760 ","End":"00:09.765","Text":"and it looks like a product,"},{"Start":"00:09.765 ","End":"00:12.735","Text":"like f of x times g of x,"},{"Start":"00:12.735 ","End":"00:17.310","Text":"and you might be thinking for derivatives, for differentiation,"},{"Start":"00:17.310 ","End":"00:20.860","Text":"there is a product rule of something times something f times g,"},{"Start":"00:20.860 ","End":"00:25.215","Text":"but unfortunately, there is no such rule for integration,"},{"Start":"00:25.215 ","End":"00:27.210","Text":"so that won\u0027t help us."},{"Start":"00:27.210 ","End":"00:30.540","Text":"Instead, what we\u0027ll do is a bit of algebra."},{"Start":"00:30.540 ","End":"00:35.170","Text":"We know how to multiply 2 terms times 2 terms."},{"Start":"00:35.170 ","End":"00:38.525","Text":"We just multiply each term from here with each term from here,"},{"Start":"00:38.525 ","End":"00:39.980","Text":"just opening the brackets,"},{"Start":"00:39.980 ","End":"00:41.300","Text":"it\u0027s what it\u0027s called."},{"Start":"00:41.300 ","End":"00:46.775","Text":"We get the integral of x squared times x is x cubed."},{"Start":"00:46.775 ","End":"00:51.075","Text":"X squared times 2 is 2x squared."},{"Start":"00:51.075 ","End":"00:54.630","Text":"1 times x is x,"},{"Start":"00:54.630 ","End":"00:57.585","Text":"and 1 times 2 is 2."},{"Start":"00:57.585 ","End":"01:03.560","Text":"I\u0027ll put it in a bracket with the dx and so now we have it in the form we want,"},{"Start":"01:03.560 ","End":"01:05.630","Text":"well, we can use this formula."},{"Start":"01:05.630 ","End":"01:09.020","Text":"Also you remember that when we have a sum of things,"},{"Start":"01:09.020 ","End":"01:11.935","Text":"we just take the sum of each separate integral."},{"Start":"01:11.935 ","End":"01:19.030","Text":"So starting with the first x cubed from the formula becomes x^4 over 4."},{"Start":"01:19.030 ","End":"01:21.675","Text":"Well, there\u0027s a plus. Plus stays."},{"Start":"01:21.675 ","End":"01:28.390","Text":"The 2 also sticks with this and x squared becomes x cubed over 3."},{"Start":"01:28.390 ","End":"01:30.695","Text":"X, which is x^1,"},{"Start":"01:30.695 ","End":"01:33.485","Text":"becomes x squared over 2,"},{"Start":"01:33.485 ","End":"01:38.015","Text":"and the constant is just a constant times 2x."},{"Start":"01:38.015 ","End":"01:39.840","Text":"Then finally there is c,"},{"Start":"01:39.840 ","End":"01:41.114","Text":"which is the constant,"},{"Start":"01:41.114 ","End":"01:44.660","Text":"and this is the answer to this 1."},{"Start":"01:44.660 ","End":"01:48.325","Text":"Let\u0027s get on to the next 1."},{"Start":"01:48.325 ","End":"01:58.850","Text":"This 1 is the integral of 1 plus 2x squared plus x^4 over x squared dx."},{"Start":"01:58.850 ","End":"02:01.160","Text":"Now once again, you might be thinking,"},{"Start":"02:01.160 ","End":"02:04.910","Text":"there might be a quotient rule, but unfortunately,"},{"Start":"02:04.910 ","End":"02:07.820","Text":"just like there\u0027s no product rule for integration,"},{"Start":"02:07.820 ","End":"02:09.895","Text":"there is 1 for differentiation,"},{"Start":"02:09.895 ","End":"02:13.490","Text":"likewise, there is a quotient rule for derivatives,"},{"Start":"02:13.490 ","End":"02:16.010","Text":"but there is no quotient rule for integrals."},{"Start":"02:16.010 ","End":"02:18.710","Text":"So again, we\u0027re going to have to use algebra,"},{"Start":"02:18.710 ","End":"02:21.620","Text":"and here the thing to do is just to take"},{"Start":"02:21.620 ","End":"02:26.225","Text":"this common denominator and divide each of the terms in the numerator by it."},{"Start":"02:26.225 ","End":"02:31.490","Text":"So what we get is if we divide everything is the integral of 1 over x"},{"Start":"02:31.490 ","End":"02:37.485","Text":"squared plus 2 x squared over x squared is just 2,"},{"Start":"02:37.485 ","End":"02:43.910","Text":"and x^4 over x squared is just x squared dx."},{"Start":"02:43.910 ","End":"02:50.000","Text":"Now, all of these we can use this formula for or the formula for a constant,"},{"Start":"02:50.000 ","End":"02:54.560","Text":"except that we still have to work a little bit more at the 1 over x squared."},{"Start":"02:54.560 ","End":"02:56.180","Text":"I\u0027ll do this at the side."},{"Start":"02:56.180 ","End":"03:01.310","Text":"1 over x squared is equal to x to the minus 2 because of"},{"Start":"03:01.310 ","End":"03:07.750","Text":"the rule that 1 over a^n is a^minus n in algebra."},{"Start":"03:07.750 ","End":"03:11.060","Text":"What I\u0027ll do is I won\u0027t copy the whole thing again,"},{"Start":"03:11.060 ","End":"03:18.850","Text":"I\u0027ll just write above it that this thing here is x^minus 2."},{"Start":"03:18.850 ","End":"03:22.110","Text":"Now we can continue with the integral."},{"Start":"03:22.110 ","End":"03:24.210","Text":"So x to the minus 2,"},{"Start":"03:24.210 ","End":"03:26.059","Text":"looking in this formula,"},{"Start":"03:26.059 ","End":"03:32.930","Text":"we raised the minus 2 by 1 and we get x^minus 1 over minus 1"},{"Start":"03:32.930 ","End":"03:41.390","Text":"plus a constant is just a constant times x and x squared becomes x^3 over 3."},{"Start":"03:41.390 ","End":"03:44.015","Text":"Again with pluses, just like the signs here,"},{"Start":"03:44.015 ","End":"03:48.220","Text":"and then plus c. If we want to,"},{"Start":"03:48.220 ","End":"03:56.365","Text":"we could write this first term as minus 1 over x plus the same everywhere else 2x plus,"},{"Start":"03:56.365 ","End":"04:03.115","Text":"maybe I\u0027ll write it as 1/3 x cubed plus c. That\u0027s it for this 1,"},{"Start":"04:03.115 ","End":"04:14.620","Text":"and now the last 1 was the integral of x plus 1 over square root of x dx."},{"Start":"04:14.620 ","End":"04:16.630","Text":"So once again, we need a trick,"},{"Start":"04:16.630 ","End":"04:22.780","Text":"and it\u0027s a trick very similar to the 1 we had here when we had a fraction,"},{"Start":"04:22.780 ","End":"04:25.450","Text":"but there was only 1 term in the denominator so"},{"Start":"04:25.450 ","End":"04:28.740","Text":"we divide each of the terms in the numerator by it."},{"Start":"04:28.740 ","End":"04:31.640","Text":"So we\u0027re doing some algebra here."},{"Start":"04:31.640 ","End":"04:37.465","Text":"So x over the square root of x is the square root of x,"},{"Start":"04:37.465 ","End":"04:40.310","Text":"and if you don\u0027t see this immediately,"},{"Start":"04:40.310 ","End":"04:44.450","Text":"then it\u0027s x^1 over x to the 1/2,"},{"Start":"04:44.450 ","End":"04:49.550","Text":"which equals x^1 minus 1/2,"},{"Start":"04:49.550 ","End":"04:56.135","Text":"which is equal x^1/2 which is again the square root of x."},{"Start":"04:56.135 ","End":"04:58.135","Text":"So that brings us here."},{"Start":"04:58.135 ","End":"05:02.900","Text":"The next term is 1 over the square root of x,"},{"Start":"05:02.900 ","End":"05:09.230","Text":"all this dx and then if we continue with algebra,"},{"Start":"05:09.230 ","End":"05:18.055","Text":"just simplifying the integral of x to the power of 1/2 and here, 1 over x^1/2,"},{"Start":"05:18.055 ","End":"05:22.405","Text":"which makes it x^1/2 dx,"},{"Start":"05:22.405 ","End":"05:25.880","Text":"and now everything is in terms of this, well,"},{"Start":"05:25.880 ","End":"05:32.480","Text":"you should have memorized by now this rule where we just raise the power by 1,"},{"Start":"05:32.480 ","End":"05:37.050","Text":"so it\u0027s x^1/2 or 3 over 2,"},{"Start":"05:37.050 ","End":"05:41.745","Text":"which is the same thing over 3 over 2 and here,"},{"Start":"05:41.745 ","End":"05:43.760","Text":"raise the power by 1,"},{"Start":"05:43.760 ","End":"05:52.385","Text":"so it becomes x^1/2 and divide by that new power and plus c of course."},{"Start":"05:52.385 ","End":"05:55.250","Text":"So this is how we could leave it,"},{"Start":"05:55.250 ","End":"06:00.770","Text":"but I prefer to continue and use the square root because that\u0027s how it was given."},{"Start":"06:00.770 ","End":"06:02.240","Text":"This is an answer,"},{"Start":"06:02.240 ","End":"06:08.075","Text":"but just a little bit of algebra will give us x times square root of x,"},{"Start":"06:08.075 ","End":"06:13.775","Text":"and the 3 over 2 in the denominator becomes 2 over 3 in the numerator."},{"Start":"06:13.775 ","End":"06:15.435","Text":"That\u0027s the first term."},{"Start":"06:15.435 ","End":"06:19.200","Text":"The second 1, dividing by 1/2 is like multiplying"},{"Start":"06:19.200 ","End":"06:23.460","Text":"by 2 and x^1/2 is the square root of x,"},{"Start":"06:23.460 ","End":"06:27.410","Text":"but still we have to carry the plus c. So this would be"},{"Start":"06:27.410 ","End":"06:34.110","Text":"a nicer way to present the answer and we\u0027re done with all 3 and so with this exercise."}],"ID":6697},{"Watched":false,"Name":"Exercise 7","Duration":"6m 58s","ChapterTopicVideoID":6639,"CourseChapterTopicPlaylistID":82571,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.com/Images/Videos_Thumbnails/6639.jpeg","UploadDate":"2022-05-16T07:37:14.8430000","DurationForVideoObject":"PT6M58S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:06.485","Text":"In this exercise, we have to compute the following integrals, 3 of them,"},{"Start":"00:06.485 ","End":"00:17.730","Text":"and we\u0027ll begin with the first integral of 4x plus 1 to the power of 10th dx,"},{"Start":"00:17.730 ","End":"00:22.050","Text":"and we have to use the right formula or tricks here."},{"Start":"00:22.050 ","End":"00:24.765","Text":"In this case, if can you look through your formula sheet,"},{"Start":"00:24.765 ","End":"00:29.280","Text":"you will find a formula that looks something like this."},{"Start":"00:29.280 ","End":"00:39.005","Text":"That the integral of ax plus b to the power of n dx"},{"Start":"00:39.005 ","End":"00:47.690","Text":"is equal to 1/a times ax plus b"},{"Start":"00:47.690 ","End":"00:57.020","Text":"to the power of n plus 1 over n plus 1 and plus c as always,"},{"Start":"00:57.020 ","End":"01:02.760","Text":"and again with the condition that n is not equal to minus 1,"},{"Start":"01:02.760 ","End":"01:05.145","Text":"which I stopped writing,"},{"Start":"01:05.145 ","End":"01:07.620","Text":"n can\u0027t be minus 1."},{"Start":"01:07.620 ","End":"01:09.830","Text":"Now we look at this and this,"},{"Start":"01:09.830 ","End":"01:12.790","Text":"and I\u0027ll use a bit of color."},{"Start":"01:12.790 ","End":"01:16.980","Text":"In this formula we take a as 4,"},{"Start":"01:16.980 ","End":"01:20.700","Text":"b as 1, and n as 10,"},{"Start":"01:20.700 ","End":"01:26.225","Text":"and then we can write the answer here as 1/ a,"},{"Start":"01:26.225 ","End":"01:30.845","Text":"meaning 1/4 times ax plus b,"},{"Start":"01:30.845 ","End":"01:37.265","Text":"which is 4x plus 1 to the power of n plus 1 over n plus 1."},{"Start":"01:37.265 ","End":"01:41.365","Text":"That\u0027s 11 and over 11,"},{"Start":"01:41.365 ","End":"01:45.015","Text":"and finally plus c,"},{"Start":"01:45.015 ","End":"01:47.900","Text":"and that\u0027s the answer."},{"Start":"01:47.900 ","End":"01:50.750","Text":"You can always simplify a bit."},{"Start":"01:50.750 ","End":"01:53.885","Text":"Put the 4 with the 11 as 44, but really,"},{"Start":"01:53.885 ","End":"01:58.460","Text":"that\u0027s now the answer to the integration."},{"Start":"01:58.460 ","End":"02:04.534","Text":"Onto the 2nd, which will also use a trick as you see."},{"Start":"02:04.534 ","End":"02:14.900","Text":"Integral of x squared minus 2x plus 1 to the power of 10 dx."},{"Start":"02:14.900 ","End":"02:20.495","Text":"Now the problem is that it looks a little bit like the previous one,"},{"Start":"02:20.495 ","End":"02:23.270","Text":"but in the previous one we had a linear term in"},{"Start":"02:23.270 ","End":"02:27.470","Text":"the brackets before the exponent 4x plus 1."},{"Start":"02:27.470 ","End":"02:29.915","Text":"Here It\u0027s a quadratic term."},{"Start":"02:29.915 ","End":"02:33.740","Text":"Well, in short, there is no such rule for this thing."},{"Start":"02:33.740 ","End":"02:36.380","Text":"We\u0027re going to have to use some trick."},{"Start":"02:36.380 ","End":"02:38.630","Text":"I\u0027m also not going to just expand it out,"},{"Start":"02:38.630 ","End":"02:40.540","Text":"which I also could if I had enough time,"},{"Start":"02:40.540 ","End":"02:43.220","Text":"I\u0027d take this thing times itself, times itself,"},{"Start":"02:43.220 ","End":"02:46.280","Text":"times itself, and even if I got clever and"},{"Start":"02:46.280 ","End":"02:49.430","Text":"figured it out to the power of 5 and then just square it,"},{"Start":"02:49.430 ","End":"02:51.095","Text":"it would be a mess."},{"Start":"02:51.095 ","End":"02:53.205","Text":"You have to just be observant,"},{"Start":"02:53.205 ","End":"02:57.945","Text":"and notice that x squared minus 2x plus 1,"},{"Start":"02:57.945 ","End":"03:00.240","Text":"and this is just in this exercise,"},{"Start":"03:00.240 ","End":"03:07.640","Text":"I\u0027m lucky that x squared minus 2x plus 1 is one of those perfect squares,"},{"Start":"03:07.640 ","End":"03:11.315","Text":"is just x minus 1 squared,"},{"Start":"03:11.315 ","End":"03:16.490","Text":"and because of this fluke or guess whoever wrote it did it deliberately,"},{"Start":"03:16.490 ","End":"03:22.730","Text":"of course, we can say that this is x minus 1 squared to the power of 10."},{"Start":"03:22.730 ","End":"03:30.040","Text":"What we get is the integral of x minus 1^20 dx,"},{"Start":"03:30.040 ","End":"03:34.700","Text":"and now we\u0027re in a very similar position as this exercise,"},{"Start":"03:34.700 ","End":"03:37.640","Text":"except that for a, b, and n,"},{"Start":"03:37.640 ","End":"03:42.160","Text":"we have different numbers and I\u0027ll write them."},{"Start":"03:42.160 ","End":"03:45.830","Text":"This time a is 1,"},{"Start":"03:45.830 ","End":"03:48.545","Text":"b is negative 1,"},{"Start":"03:48.545 ","End":"03:50.375","Text":"and n is 20."},{"Start":"03:50.375 ","End":"03:53.225","Text":"Using this same formula again,"},{"Start":"03:53.225 ","End":"03:57.665","Text":"we get that this thing is equal to 1/a,"},{"Start":"03:57.665 ","End":"04:05.410","Text":"which is 1/1 and ax plus b is the x minus 1."},{"Start":"04:05.410 ","End":"04:10.670","Text":"Notice I added this 1 here just so we\u0027d have something to say what a was."},{"Start":"04:10.670 ","End":"04:12.095","Text":"It would be a coefficient,"},{"Start":"04:12.095 ","End":"04:17.345","Text":"so this is x minus 1 to the power of n plus 1,"},{"Start":"04:17.345 ","End":"04:23.660","Text":"this time is 21 and over 21 plus C,"},{"Start":"04:23.660 ","End":"04:26.450","Text":"and that\u0027s the answer to this one."},{"Start":"04:26.450 ","End":"04:29.525","Text":"Then we come to the last one,"},{"Start":"04:29.525 ","End":"04:38.495","Text":"which is the integral of 4/x minus 2^5 dx of course,"},{"Start":"04:38.495 ","End":"04:44.330","Text":"what we\u0027re going to do is basically use our standard formula,"},{"Start":"04:44.330 ","End":"04:48.860","Text":"this one, and once again, I\u0027ll use colors,"},{"Start":"04:48.860 ","End":"04:50.300","Text":"but before I do that,"},{"Start":"04:50.300 ","End":"04:53.520","Text":"I need to get it into this form."},{"Start":"04:53.520 ","End":"04:56.240","Text":"This is equal to 4."},{"Start":"04:56.240 ","End":"05:05.240","Text":"We can put the 4 aside and then write it as x minus 2 to the power of negative 5,"},{"Start":"05:05.240 ","End":"05:08.780","Text":"like we\u0027ve always been doing, dx,"},{"Start":"05:08.780 ","End":"05:11.795","Text":"and then with the coloring,"},{"Start":"05:11.795 ","End":"05:15.070","Text":"we see that the a is 1."},{"Start":"05:15.070 ","End":"05:16.550","Text":"There wasn\u0027t anything here,"},{"Start":"05:16.550 ","End":"05:17.690","Text":"but I put a 1 in,"},{"Start":"05:17.690 ","End":"05:19.370","Text":"it was just like x minus 2,"},{"Start":"05:19.370 ","End":"05:21.350","Text":"but that\u0027s 1x minus 2,"},{"Start":"05:21.350 ","End":"05:24.380","Text":"and the green, which is b,"},{"Start":"05:24.380 ","End":"05:30.795","Text":"is minus 2, and the exponent n is minus 5."},{"Start":"05:30.795 ","End":"05:35.760","Text":"Using this formula, the 4 stays as 4."},{"Start":"05:35.760 ","End":"05:40.130","Text":"We said that that\u0027s one rule about integration that a constant times something,"},{"Start":"05:40.130 ","End":"05:42.110","Text":"the constant just stays there,"},{"Start":"05:42.110 ","End":"05:44.875","Text":"and now we can use this rule."},{"Start":"05:44.875 ","End":"05:46.845","Text":"The 4 has brackets."},{"Start":"05:46.845 ","End":"05:50.210","Text":"Now 1/a and a was 1,"},{"Start":"05:50.210 ","End":"05:53.435","Text":"so that\u0027s just 1 so I don\u0027t need to bother writing it,"},{"Start":"05:53.435 ","End":"05:56.045","Text":"and I get the ax plus b part,"},{"Start":"05:56.045 ","End":"06:06.449","Text":"which is x minus 2 to the power of minus 5 plus 1 is minus 4 over minus 4,"},{"Start":"06:06.449 ","End":"06:09.195","Text":"and at the end plus C,"},{"Start":"06:09.195 ","End":"06:11.900","Text":"and we just need to, we don\u0027t have to."},{"Start":"06:11.900 ","End":"06:13.820","Text":"This is the end of the integration,"},{"Start":"06:13.820 ","End":"06:18.665","Text":"but I\u0027d like to simplify it and say that this is"},{"Start":"06:18.665 ","End":"06:24.075","Text":"equal to 4 over minus 4 is minus 1."},{"Start":"06:24.075 ","End":"06:30.680","Text":"I can write it as minus and then x minus 2 to the minus 4 using"},{"Start":"06:30.680 ","End":"06:37.565","Text":"the usual formula that a to the minus n is 1/a^n."},{"Start":"06:37.565 ","End":"06:43.990","Text":"This is equal to 1 over x minus 2 to the power of 4,"},{"Start":"06:43.990 ","End":"06:46.400","Text":"and plus C. It\u0027s like"},{"Start":"06:46.400 ","End":"06:50.510","Text":"simplification simply because there wasn\u0027t any negative exponents here,"},{"Start":"06:50.510 ","End":"06:52.850","Text":"I kept it in the same format,"},{"Start":"06:52.850 ","End":"06:59.160","Text":"and that\u0027s it for the 3rd one and to the exercise."}],"ID":6698},{"Watched":false,"Name":"Exercise 8","Duration":"8m 7s","ChapterTopicVideoID":6640,"CourseChapterTopicPlaylistID":82571,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.com/Images/Videos_Thumbnails/6640.jpeg","UploadDate":"2016-07-19T11:48:24.8700000","DurationForVideoObject":"PT8M7S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:03.750","Text":"In this clip we have to compute the following integrals."},{"Start":"00:03.750 ","End":"00:07.860","Text":"They are a, b and c. Let\u0027s start with a, I\u0027ve copied it already."},{"Start":"00:07.860 ","End":"00:11.295","Text":"The cube root of 4x minus 10."},{"Start":"00:11.295 ","End":"00:15.030","Text":"At the side I\u0027ve written the formula that we\u0027ve been using lately."},{"Start":"00:15.030 ","End":"00:17.755","Text":"It probably will help us again here."},{"Start":"00:17.755 ","End":"00:22.955","Text":"Here what I need is to get something that looks like the left hand side here,"},{"Start":"00:22.955 ","End":"00:24.980","Text":"something of this form."},{"Start":"00:24.980 ","End":"00:27.680","Text":"Now here I have to the power of n and here I have"},{"Start":"00:27.680 ","End":"00:31.100","Text":"a cube root but from algebra that\u0027s easily convertible."},{"Start":"00:31.100 ","End":"00:33.565","Text":"A root is convertible to an exponent."},{"Start":"00:33.565 ","End":"00:39.915","Text":"If I just write this as 4x minus 10 to the power of 1/3 then I\u0027ve got the same thing."},{"Start":"00:39.915 ","End":"00:41.660","Text":"But it looks like this form,"},{"Start":"00:41.660 ","End":"00:43.690","Text":"and I\u0027ll just color it to show you."},{"Start":"00:43.690 ","End":"00:45.720","Text":"We see that a is 4,"},{"Start":"00:45.720 ","End":"00:48.420","Text":"b is minus 10, n is 1/3."},{"Start":"00:48.420 ","End":"00:52.460","Text":"If I use the right hand side I can get what the integral is."},{"Start":"00:52.460 ","End":"00:54.980","Text":"The integral is 1 over a,"},{"Start":"00:54.980 ","End":"01:01.590","Text":"which is 1/4 times 4x minus 10 to the power of n plus 1,"},{"Start":"01:01.590 ","End":"01:09.080","Text":"1/3 plus 1 is 4/3 over 4/3 and plus c. That\u0027s the answer."},{"Start":"01:09.080 ","End":"01:12.440","Text":"But just to be nice we\u0027ll simplify it and"},{"Start":"01:12.440 ","End":"01:16.700","Text":"return it to the form of the root rather than the exponent."},{"Start":"01:16.700 ","End":"01:21.890","Text":"What we\u0027ll get is if I divide by 4/3 is like multiplying by 3/4,"},{"Start":"01:21.890 ","End":"01:25.390","Text":"so this gives us 3/16."},{"Start":"01:25.390 ","End":"01:27.840","Text":"Something to the power of 4/3,"},{"Start":"01:27.840 ","End":"01:33.220","Text":"like a to the power of 4 over 3 is a to the power of 1 and 1/3,"},{"Start":"01:33.220 ","End":"01:37.085","Text":"which is a times a to the power of 1/3."},{"Start":"01:37.085 ","End":"01:41.105","Text":"That is to simply equal to a cube root of a."},{"Start":"01:41.105 ","End":"01:45.275","Text":"Using that I can write this as 3/16 times"},{"Start":"01:45.275 ","End":"01:52.520","Text":"4x minus 10 times the cube root of 4x minus 10."},{"Start":"01:52.520 ","End":"01:57.850","Text":"Or another way of doing it just alternatively though it doesn\u0027t matter,"},{"Start":"01:57.850 ","End":"02:00.035","Text":"I\u0027ll write it for you, but I won\u0027t explain,"},{"Start":"02:00.035 ","End":"02:10.085","Text":"you could also take it as 3/16 times the cube root of 4x minus 10 to the power of 4."},{"Start":"02:10.085 ","End":"02:12.680","Text":"But I don\u0027t want to get into too much detail since this"},{"Start":"02:12.680 ","End":"02:15.020","Text":"is adequate but it could go this way or it"},{"Start":"02:15.020 ","End":"02:20.440","Text":"could go this way when you simplify it. Part b."},{"Start":"02:20.440 ","End":"02:23.175","Text":"Again, I want to use this formula."},{"Start":"02:23.175 ","End":"02:25.765","Text":"What I need to do is a bit of algebra."},{"Start":"02:25.765 ","End":"02:30.935","Text":"This time I\u0027ll do a little bit quicker because we\u0027ve seen this thing many times."},{"Start":"02:30.935 ","End":"02:35.330","Text":"What I\u0027ll do is that this is the integral of 10 and 1 over"},{"Start":"02:35.330 ","End":"02:39.770","Text":"the square root of something is that something to the power of minus 1/2."},{"Start":"02:39.770 ","End":"02:43.450","Text":"The 1/2 for the square root and the minus because it\u0027s in the denominator,"},{"Start":"02:43.450 ","End":"02:49.260","Text":"so it\u0027s 2x plus 4 to the power of minus 1/2 dx."},{"Start":"02:49.260 ","End":"02:51.850","Text":"At this point I can use this formula."},{"Start":"02:51.850 ","End":"02:54.530","Text":"I just have to remember that also that if I have a"},{"Start":"02:54.530 ","End":"02:57.260","Text":"constant the constant just stays there."},{"Start":"02:57.260 ","End":"03:00.545","Text":"The constant multiplied by something just stays as 10."},{"Start":"03:00.545 ","End":"03:02.860","Text":"Then now I use this formula."},{"Start":"03:02.860 ","End":"03:08.570","Text":"Again, I\u0027m not going to do the coloring because you can see that a is 2 and b is 4,"},{"Start":"03:08.570 ","End":"03:10.190","Text":"and n is minus 1/2."},{"Start":"03:10.190 ","End":"03:12.620","Text":"What I get is 1 over a,"},{"Start":"03:12.620 ","End":"03:14.739","Text":"which is 1 over 2."},{"Start":"03:14.739 ","End":"03:18.200","Text":"Then, I get this thing to the power of n plus 1,"},{"Start":"03:18.200 ","End":"03:19.865","Text":"whatever was in the brackets there,"},{"Start":"03:19.865 ","End":"03:22.555","Text":"2x plus 4, I add 1,"},{"Start":"03:22.555 ","End":"03:28.520","Text":"so instead of minus 1/2 it\u0027ll be plus 1/2 and then divide it by that exponent,"},{"Start":"03:28.520 ","End":"03:34.685","Text":"which is over 1/2 and always the plus c because it\u0027s an indefinite integral."},{"Start":"03:34.685 ","End":"03:37.895","Text":"That\u0027s the answer but we could simplify."},{"Start":"03:37.895 ","End":"03:43.305","Text":"For example, dividing by 1/2 is like multiplying by 2 so this goes with this."},{"Start":"03:43.305 ","End":"03:45.645","Text":"We\u0027re left with 10."},{"Start":"03:45.645 ","End":"03:48.650","Text":"Then the 1/2 can be written as a square root,"},{"Start":"03:48.650 ","End":"03:51.810","Text":"square root of 2x plus 4 plus"},{"Start":"03:51.810 ","End":"03:56.685","Text":"c. It looks a bit tidier than what we had written over here."},{"Start":"03:56.685 ","End":"03:58.985","Text":"That\u0027s the answer to part b."},{"Start":"03:58.985 ","End":"04:02.330","Text":"Now let\u0027s get on to part c,"},{"Start":"04:02.330 ","End":"04:05.370","Text":"where we definitely will have a trick."},{"Start":"04:05.370 ","End":"04:07.555","Text":"I\u0027ll write down what was c,"},{"Start":"04:07.555 ","End":"04:16.460","Text":"which is the integral of x over x minus 1 to the power of 4 dx."},{"Start":"04:16.460 ","End":"04:19.715","Text":"Here\u0027s where we\u0027re temporarily stuck."},{"Start":"04:19.715 ","End":"04:23.750","Text":"Because it would be nice to see this as a fraction if we"},{"Start":"04:23.750 ","End":"04:27.720","Text":"had a rule for a fraction for f over g. All ready mentioned,"},{"Start":"04:27.720 ","End":"04:32.495","Text":"there is no product rule and there is no quotient rule in integrals?"},{"Start":"04:32.495 ","End":"04:36.350","Text":"What we have to do is think of some clever way out of this."},{"Start":"04:36.350 ","End":"04:42.185","Text":"After pondering it a bit I realized that if I had instead of x,"},{"Start":"04:42.185 ","End":"04:43.895","Text":"x minus 1 here,"},{"Start":"04:43.895 ","End":"04:46.055","Text":"that would be very good for me."},{"Start":"04:46.055 ","End":"04:51.500","Text":"Because then x minus 1 would cancel with 1 of the x minus 1 here and I"},{"Start":"04:51.500 ","End":"04:57.175","Text":"already know how to do 1 over x minus 1 cubed and get it into this form."},{"Start":"04:57.175 ","End":"04:59.595","Text":"Let me start by writing something."},{"Start":"04:59.595 ","End":"05:02.990","Text":"I\u0027ll write the the same x minus 1 to the 4th here,"},{"Start":"05:02.990 ","End":"05:06.530","Text":"by wishful thinking I wish I had x minus 1"},{"Start":"05:06.530 ","End":"05:11.150","Text":"here but I don\u0027t because I\u0027ve changed the exercise."},{"Start":"05:11.150 ","End":"05:15.750","Text":"But I can\u0027t compensate if I add plus 1 here,"},{"Start":"05:15.750 ","End":"05:19.370","Text":"and I haven\u0027t changed the exercise because minus 1 plus 1 is okay."},{"Start":"05:19.370 ","End":"05:21.980","Text":"I could put a bracket around here also."},{"Start":"05:21.980 ","End":"05:23.885","Text":"Now how does this help me?"},{"Start":"05:23.885 ","End":"05:26.630","Text":"Because I can use fractions."},{"Start":"05:26.630 ","End":"05:29.180","Text":"When I have several terms in the numerator I can"},{"Start":"05:29.180 ","End":"05:32.390","Text":"take each 1 separately over the denominator."},{"Start":"05:32.390 ","End":"05:36.095","Text":"This is equal to integral of the sum of 2 things."},{"Start":"05:36.095 ","End":"05:43.260","Text":"On the 1 hand, I mean, part 1 is x minus 1 over x minus 1 to the 4th,"},{"Start":"05:43.260 ","End":"05:52.250","Text":"and plus the other part which is 1 over x minus 1 to the 4th and all this, dx."},{"Start":"05:52.250 ","End":"05:54.350","Text":"Now here we have the integral,"},{"Start":"05:54.350 ","End":"05:55.990","Text":"the sum of 2 things."},{"Start":"05:55.990 ","End":"06:02.520","Text":"The first 1, if I cancel an x minus 1 all I\u0027m left with is 1 and here I\u0027m left with 3."},{"Start":"06:02.520 ","End":"06:12.375","Text":"I have 1 over x minus 1 cubed plus 1 over x minus 1 to the 4th."},{"Start":"06:12.375 ","End":"06:15.410","Text":"All this, dx."},{"Start":"06:15.410 ","End":"06:20.435","Text":"Now on each of these pieces I can use this rule because,"},{"Start":"06:20.435 ","End":"06:24.440","Text":"well, I\u0027ll just do 1 more step here then it will be immediately obvious."},{"Start":"06:24.440 ","End":"06:30.080","Text":"It\u0027s the integral of x minus 1 to the power of"},{"Start":"06:30.080 ","End":"06:37.305","Text":"minus 3 and these 2 bits separately, dx."},{"Start":"06:37.305 ","End":"06:39.615","Text":"Now, each 1 of these we\u0027re ready,"},{"Start":"06:39.615 ","End":"06:43.070","Text":"we\u0027ve so practiced with this rule here"},{"Start":"06:43.070 ","End":"06:46.505","Text":"that will just write the answer almost straight away."},{"Start":"06:46.505 ","End":"06:54.330","Text":"Obviously the a is 1 and the b is minus 1 and 1 time will take n as minus 3,"},{"Start":"06:54.330 ","End":"06:57.605","Text":"and 1 time will take n as being minus 4."},{"Start":"06:57.605 ","End":"07:01.430","Text":"What we get is the 1 over a just becomes 1."},{"Start":"07:01.430 ","End":"07:07.460","Text":"For this bit we get x minus 1 and we raise the power by 1,"},{"Start":"07:07.460 ","End":"07:11.375","Text":"which is minus 2 over minus 2."},{"Start":"07:11.375 ","End":"07:15.960","Text":"Then the other bit we also get the 1 over a is just 1."},{"Start":"07:15.960 ","End":"07:24.600","Text":"We get x minus 1 and raised by 1 is to the minus 3 over minus 3."},{"Start":"07:24.600 ","End":"07:26.925","Text":"This is the answer."},{"Start":"07:26.925 ","End":"07:32.630","Text":"But if we wanted to tidy it up and not use negative exponents,"},{"Start":"07:32.630 ","End":"07:36.680","Text":"then what we could do is finally write it as,"},{"Start":"07:36.680 ","End":"07:39.830","Text":"and I\u0027ll put maybe the minus 1/2 separately."},{"Start":"07:39.830 ","End":"07:46.020","Text":"It\u0027s minus 1/2 times 1 over x minus 1 squared"},{"Start":"07:46.020 ","End":"07:55.325","Text":"minus 1/3 of 1 over x minus 1 to the power of 3 and plus c,"},{"Start":"07:55.325 ","End":"07:59.260","Text":"which I also forgot to put over here."},{"Start":"07:59.260 ","End":"08:02.630","Text":"That\u0027s about as far as we can go with this."},{"Start":"08:02.630 ","End":"08:07.890","Text":"That\u0027s the answer. On to the next."}],"ID":6699},{"Watched":false,"Name":"Exercise 9","Duration":"12m 20s","ChapterTopicVideoID":6641,"CourseChapterTopicPlaylistID":82571,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.com/Images/Videos_Thumbnails/6641.jpeg","UploadDate":"2016-07-19T11:49:39.7200000","DurationForVideoObject":"PT12M20S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:03.930","Text":"In this exercise, I have to compute the following integrals,"},{"Start":"00:03.930 ","End":"00:05.535","Text":"and there are 3 of them."},{"Start":"00:05.535 ","End":"00:14.790","Text":"Let\u0027s start with the 1st 1 which is the integral of dx over square root of x minus 1,"},{"Start":"00:14.790 ","End":"00:17.400","Text":"minus the square root of x."},{"Start":"00:17.400 ","End":"00:23.850","Text":"Here, I run into a bit of difficulty because I don\u0027t really know how to tackle this 1."},{"Start":"00:23.850 ","End":"00:30.825","Text":"I don\u0027t have a formula for the integral of f over g quotient,"},{"Start":"00:30.825 ","End":"00:32.295","Text":"there is no such formula."},{"Start":"00:32.295 ","End":"00:35.459","Text":"There is for derivatives but not for integrals."},{"Start":"00:35.459 ","End":"00:39.675","Text":"I don\u0027t even have formula for the integral of 1 over something,"},{"Start":"00:39.675 ","End":"00:41.840","Text":"that\u0027s also not going to help me."},{"Start":"00:41.840 ","End":"00:45.080","Text":"I appear to be stuck because I have a quotient and"},{"Start":"00:45.080 ","End":"00:48.560","Text":"I\u0027d like to somehow convert that into a sum or a difference,"},{"Start":"00:48.560 ","End":"00:50.620","Text":"or something I know how to deal with."},{"Start":"00:50.620 ","End":"00:53.780","Text":"But when I look closely at the denominator,"},{"Start":"00:53.780 ","End":"00:55.835","Text":"it really reminds me of something."},{"Start":"00:55.835 ","End":"00:59.810","Text":"I see the square root of something minus the square root of something,"},{"Start":"00:59.810 ","End":"01:03.260","Text":"and it reminds me of what is called the conjugate."},{"Start":"01:03.260 ","End":"01:11.900","Text":"Remember that the conjugate of anything of the form a minus b is just a plus b,"},{"Start":"01:11.900 ","End":"01:13.310","Text":"and the other way around,"},{"Start":"01:13.310 ","End":"01:15.580","Text":"the conjugate of a plus b is a minus b,"},{"Start":"01:15.580 ","End":"01:18.484","Text":"and if you multiply something by its conjugate,"},{"Start":"01:18.484 ","End":"01:25.150","Text":"what you get according to the difference of squares formula is a squared minus b squared."},{"Start":"01:25.150 ","End":"01:30.530","Text":"Now, this is very good for us because if a or b or both have a square root over them,"},{"Start":"01:30.530 ","End":"01:32.120","Text":"then when we square it,"},{"Start":"01:32.120 ","End":"01:34.235","Text":"we get rid of the square root."},{"Start":"01:34.235 ","End":"01:40.790","Text":"Let\u0027s try multiplying this thing by its conjugate and see what we get."},{"Start":"01:40.790 ","End":"01:48.320","Text":"If we take the square root of x minus 1 minus the square root of x,"},{"Start":"01:48.320 ","End":"01:50.610","Text":"which is our a minus b,"},{"Start":"01:50.610 ","End":"01:53.060","Text":"and multiply it by its conjugate,"},{"Start":"01:53.060 ","End":"01:58.310","Text":"which is the square root of x minus 1 plus the square root of x."},{"Start":"01:58.310 ","End":"02:01.055","Text":"Then according to this formula,"},{"Start":"02:01.055 ","End":"02:02.720","Text":"we get a squared."},{"Start":"02:02.720 ","End":"02:07.295","Text":"The a squared is just going to be x minus 1 without the square root."},{"Start":"02:07.295 ","End":"02:09.755","Text":"When I take the square root of x squared,"},{"Start":"02:09.755 ","End":"02:13.129","Text":"it\u0027s just going to be x with a minus,"},{"Start":"02:13.129 ","End":"02:14.730","Text":"without the square root,"},{"Start":"02:14.730 ","End":"02:18.095","Text":"and we end up with just minus 1,"},{"Start":"02:18.095 ","End":"02:22.870","Text":"which is a very simple quantity and easy to deal with."},{"Start":"02:22.870 ","End":"02:27.850","Text":"Let\u0027s somehow try and multiply this denominator by the conjugate."},{"Start":"02:27.850 ","End":"02:31.230","Text":"Let me start off with something I think it\u0027s not quite right,"},{"Start":"02:31.230 ","End":"02:33.015","Text":"and then we\u0027ll put it right."},{"Start":"02:33.015 ","End":"02:37.385","Text":"Let\u0027s try multiplying this dx"},{"Start":"02:37.385 ","End":"02:43.160","Text":"over the square root of x minus 1 minus the square root of x,"},{"Start":"02:43.160 ","End":"02:46.400","Text":"and let\u0027s multiply the denominator by"},{"Start":"02:46.400 ","End":"02:51.575","Text":"the square root of x minus 1 plus the square root of x."},{"Start":"02:51.575 ","End":"02:56.580","Text":"However, you can\u0027t just multiply the denominator by something,"},{"Start":"02:56.580 ","End":"02:58.310","Text":"I mean, I\u0027ve changed the exercise."},{"Start":"02:58.310 ","End":"03:02.515","Text":"But if to be fair I multiply the numerator also,"},{"Start":"03:02.515 ","End":"03:06.320","Text":"then there\u0027s no problem because I have multiplied by 1."},{"Start":"03:06.320 ","End":"03:08.510","Text":"If you multiply by something over itself,"},{"Start":"03:08.510 ","End":"03:10.805","Text":"that\u0027s just 1, and now it\u0027s fair."},{"Start":"03:10.805 ","End":"03:14.560","Text":"Now we\u0027ll multiply the denominators first,"},{"Start":"03:14.560 ","End":"03:19.460","Text":"but we\u0027ve already done this and their product is minus 1."},{"Start":"03:19.460 ","End":"03:22.130","Text":"But instead of writing minus 1 in the denominator,"},{"Start":"03:22.130 ","End":"03:24.515","Text":"I\u0027ll just multiply the whole thing by minus."},{"Start":"03:24.515 ","End":"03:26.610","Text":"I can put the minus anywhere basically,"},{"Start":"03:26.610 ","End":"03:29.365","Text":"and I could put the minus in front of the integral,"},{"Start":"03:29.365 ","End":"03:33.530","Text":"and now I can get rid of this because I\u0027ve just used the minus 1,"},{"Start":"03:33.530 ","End":"03:36.515","Text":"and what we\u0027re left with is just the numerator,"},{"Start":"03:36.515 ","End":"03:44.650","Text":"which is the square root of x minus 1 plus the square root of x, dx."},{"Start":"03:44.650 ","End":"03:48.290","Text":"This is great because further this nuisance,"},{"Start":"03:48.290 ","End":"03:51.470","Text":"some dividing sign with the quotient,"},{"Start":"03:51.470 ","End":"03:53.140","Text":"we now have a plus,"},{"Start":"03:53.140 ","End":"03:55.640","Text":"and that\u0027s much better for us because we know there is"},{"Start":"03:55.640 ","End":"03:58.805","Text":"a formula for the sum of integrals."},{"Start":"03:58.805 ","End":"04:01.925","Text":"Now, what we\u0027re going to need here is just a little formula,"},{"Start":"04:01.925 ","End":"04:09.260","Text":"and the formula that I\u0027m going to write is that ax plus b to the power of n,"},{"Start":"04:09.260 ","End":"04:18.200","Text":"dx is equal to 1 over a times ax plus b to the power of n plus 1,"},{"Start":"04:18.200 ","End":"04:23.885","Text":"over n plus 1 plus c. But in our case,"},{"Start":"04:23.885 ","End":"04:26.420","Text":"a is 1 in each of these,"},{"Start":"04:26.420 ","End":"04:31.310","Text":"because I\u0027m going to make this square root as something to the power of a 1/2,"},{"Start":"04:31.310 ","End":"04:34.400","Text":"and the coefficient of x in each case is going to be 1."},{"Start":"04:34.400 ","End":"04:43.870","Text":"If a is 1, this formula really simplifies and we get that the integral of x plus b,"},{"Start":"04:43.870 ","End":"04:46.080","Text":"put it in brackets,"},{"Start":"04:46.080 ","End":"04:49.590","Text":"to the power of n dx is simply,"},{"Start":"04:49.590 ","End":"04:51.450","Text":"now here is 1, so we don\u0027t need it,"},{"Start":"04:51.450 ","End":"04:56.175","Text":"so it\u0027s just x plus b to the power of n plus 1,"},{"Start":"04:56.175 ","End":"04:58.240","Text":"over n plus 1,"},{"Start":"04:58.240 ","End":"05:00.830","Text":"plus c, almost as if it was x,"},{"Start":"05:00.830 ","End":"05:02.810","Text":"but it\u0027s x plus b."},{"Start":"05:02.810 ","End":"05:08.570","Text":"We\u0027ll continue. This equals minus the integral."},{"Start":"05:08.570 ","End":"05:11.110","Text":"Note, square root is to the power of 1/2,"},{"Start":"05:11.110 ","End":"05:14.795","Text":"so we have x minus 1 to the power of a 1/2,"},{"Start":"05:14.795 ","End":"05:17.705","Text":"plus x to the power of a 1/2,"},{"Start":"05:17.705 ","End":"05:20.510","Text":"and all this dx."},{"Start":"05:20.510 ","End":"05:22.640","Text":"Now, because we have a sum,"},{"Start":"05:22.640 ","End":"05:27.410","Text":"there is a formula for integration of a sum and we just add each 1 separately,"},{"Start":"05:27.410 ","End":"05:28.970","Text":"so this is minus."},{"Start":"05:28.970 ","End":"05:31.684","Text":"Now, x minus 1 to the 1/2, the integral,"},{"Start":"05:31.684 ","End":"05:35.500","Text":"I use this formula with b equals minus 1."},{"Start":"05:35.500 ","End":"05:38.400","Text":"I get x minus 1."},{"Start":"05:38.400 ","End":"05:40.965","Text":"Now, instead of a 1/2, a 1/2 plus 1,"},{"Start":"05:40.965 ","End":"05:42.885","Text":"which is 3 over 2,"},{"Start":"05:42.885 ","End":"05:45.480","Text":"over 3 over 2."},{"Start":"05:45.480 ","End":"05:49.680","Text":"Here, similarly, just raise the 1/2 by 1,"},{"Start":"05:49.680 ","End":"05:52.530","Text":"so it\u0027s x to the 3 over 2,"},{"Start":"05:52.530 ","End":"05:55.215","Text":"and divide by 3 over 2,"},{"Start":"05:55.215 ","End":"05:59.080","Text":"and all this plus c. In fact,"},{"Start":"05:59.080 ","End":"06:00.430","Text":"we can simplify this."},{"Start":"06:00.430 ","End":"06:06.685","Text":"We could take the 3 over 2 outside the brackets and even put it as 2/3 and the numerator."},{"Start":"06:06.685 ","End":"06:11.530","Text":"But we won\u0027t continue with this because this already is the answer to the integration,"},{"Start":"06:11.530 ","End":"06:14.110","Text":"and the rest of it is just simplification."},{"Start":"06:14.110 ","End":"06:17.770","Text":"We\u0027re done, and that was nice use of the conjugate,"},{"Start":"06:17.770 ","End":"06:20.750","Text":"which we\u0027ll use again in the future."},{"Start":"06:20.750 ","End":"06:22.725","Text":"That\u0027s it with part a,"},{"Start":"06:22.725 ","End":"06:25.110","Text":"now on to part b."},{"Start":"06:25.110 ","End":"06:28.115","Text":"What we have is this here,"},{"Start":"06:28.115 ","End":"06:32.855","Text":"x over the square root of x plus 1, plus 1, dx."},{"Start":"06:32.855 ","End":"06:35.075","Text":"We\u0027re going to use the same tricks as before."},{"Start":"06:35.075 ","End":"06:37.655","Text":"First of all the trick of the conjugate."},{"Start":"06:37.655 ","End":"06:45.010","Text":"We\u0027ll take this and we\u0027ll multiply top and bottom by the conjugate of the denominator."},{"Start":"06:45.010 ","End":"06:51.230","Text":"We\u0027ll take a dividing line here and put the square root of x plus 1."},{"Start":"06:51.230 ","End":"06:52.399","Text":"Since it\u0027s the conjugate,"},{"Start":"06:52.399 ","End":"06:54.665","Text":"now we have minus 1."},{"Start":"06:54.665 ","End":"06:56.210","Text":"As not to change anything,"},{"Start":"06:56.210 ","End":"07:00.320","Text":"we multiply the numerator by the same thing as the denominator,"},{"Start":"07:00.320 ","End":"07:03.385","Text":"so x plus 1 minus 1."},{"Start":"07:03.385 ","End":"07:05.700","Text":"Now, when we multiply this out,"},{"Start":"07:05.700 ","End":"07:09.925","Text":"on the bottom we get the product of conjugates,"},{"Start":"07:09.925 ","End":"07:15.110","Text":"and just to remind you that a plus b times"},{"Start":"07:15.110 ","End":"07:20.675","Text":"a minus b is a squared minus b squared."},{"Start":"07:20.675 ","End":"07:23.300","Text":"Then on the denominator,"},{"Start":"07:23.300 ","End":"07:27.500","Text":"we get x plus 1 square root, all squared,"},{"Start":"07:27.500 ","End":"07:29.830","Text":"which is just x plus 1,"},{"Start":"07:29.830 ","End":"07:32.550","Text":"and then minus the b squared,"},{"Start":"07:32.550 ","End":"07:35.145","Text":"which is 1 squared, which is 1."},{"Start":"07:35.145 ","End":"07:41.825","Text":"On the numerator, we have the x times this difference,"},{"Start":"07:41.825 ","End":"07:46.955","Text":"the square root of x plus 1 minus 1,"},{"Start":"07:46.955 ","End":"07:49.205","Text":"and if we simplify it,"},{"Start":"07:49.205 ","End":"07:53.535","Text":"then we will get x plus 1 minus 1,"},{"Start":"07:53.535 ","End":"07:56.640","Text":"all this here is just x,"},{"Start":"07:56.640 ","End":"07:58.200","Text":"like plus 1 minus 1,"},{"Start":"07:58.200 ","End":"08:01.125","Text":"and x cancels with x."},{"Start":"08:01.125 ","End":"08:03.410","Text":"After all this cancellation,"},{"Start":"08:03.410 ","End":"08:10.730","Text":"we just get the integral of the square root of x plus 1 minus 1,"},{"Start":"08:10.730 ","End":"08:12.905","Text":"and all this dx."},{"Start":"08:12.905 ","End":"08:15.440","Text":"Hope you followed that. X plus 1 minus 1 is x,"},{"Start":"08:15.440 ","End":"08:18.395","Text":"x cancels with x, and we\u0027re left with just this."},{"Start":"08:18.395 ","End":"08:22.505","Text":"Then using exponents instead of square roots,"},{"Start":"08:22.505 ","End":"08:31.040","Text":"we get x plus 1 to the power of 1/2 minus 1, all this dx."},{"Start":"08:31.040 ","End":"08:36.570","Text":"This is a difference and set of the quotient that we had before,"},{"Start":"08:36.570 ","End":"08:38.180","Text":"and this we know how to deal with."},{"Start":"08:38.180 ","End":"08:43.025","Text":"We take the integral of this separately minus the integral of this separately,"},{"Start":"08:43.025 ","End":"08:46.450","Text":"and we\u0027re going to use the same formula as we did in the previous exercise,"},{"Start":"08:46.450 ","End":"08:47.475","Text":"and I won\u0027t repeat it,"},{"Start":"08:47.475 ","End":"08:54.820","Text":"with x plus b to the power of n. What we get is x plus b, which is the x plus 1,"},{"Start":"08:54.820 ","End":"08:57.410","Text":"to the power of n plus 1,"},{"Start":"08:57.410 ","End":"08:59.930","Text":"which in this case is 3 over 2,"},{"Start":"08:59.930 ","End":"09:02.540","Text":"also over 3 over 2,"},{"Start":"09:02.540 ","End":"09:04.715","Text":"minus, so it\u0027s minus."},{"Start":"09:04.715 ","End":"09:07.160","Text":"The integral of a is just ax,"},{"Start":"09:07.160 ","End":"09:09.245","Text":"so we put minus 1x,"},{"Start":"09:09.245 ","End":"09:12.635","Text":"and minus 1x is just x,"},{"Start":"09:12.635 ","End":"09:15.920","Text":"and all this plus c. Of course we could"},{"Start":"09:15.920 ","End":"09:21.360","Text":"simplify and we could put the 3 over 2 as 2 over 3 in the numerator,"},{"Start":"09:21.360 ","End":"09:24.695","Text":"the 3 over 2 could be using the square root of x."},{"Start":"09:24.695 ","End":"09:28.130","Text":"But this is the integration part,"},{"Start":"09:28.130 ","End":"09:30.710","Text":"and I\u0027m not going to continue with the simplification,"},{"Start":"09:30.710 ","End":"09:33.295","Text":"and this is the answer to part b."},{"Start":"09:33.295 ","End":"09:35.965","Text":"Next we come to part c,"},{"Start":"09:35.965 ","End":"09:38.025","Text":"and I\u0027ve rewritten it over here."},{"Start":"09:38.025 ","End":"09:41.840","Text":"We\u0027re going to use the same tricks as we used in a and b,"},{"Start":"09:41.840 ","End":"09:46.075","Text":"which is mostly the conjugate and certain integration rules."},{"Start":"09:46.075 ","End":"09:50.915","Text":"Let\u0027s start with multiplying top and bottom by the conjugate because we see"},{"Start":"09:50.915 ","End":"09:55.835","Text":"on the bottom we have the square root of something and there\u0027s a plus or minus."},{"Start":"09:55.835 ","End":"09:58.699","Text":"With every indication we should use the conjugate,"},{"Start":"09:58.699 ","End":"10:01.160","Text":"so it\u0027s multiply the top and bottom."},{"Start":"10:01.160 ","End":"10:03.500","Text":"Here we have the conjugate to this,"},{"Start":"10:03.500 ","End":"10:06.275","Text":"which is the square root of x plus 1."},{"Start":"10:06.275 ","End":"10:08.420","Text":"Just to make it right and to be fair,"},{"Start":"10:08.420 ","End":"10:11.705","Text":"we have to multiply the top also by the same thing,"},{"Start":"10:11.705 ","End":"10:16.165","Text":"square root of x plus 1, and the dx."},{"Start":"10:16.165 ","End":"10:20.010","Text":"Next thing to do is to multiply the conjugates."},{"Start":"10:20.010 ","End":"10:22.855","Text":"As usual, a minus b,"},{"Start":"10:22.855 ","End":"10:27.905","Text":"a plus b, is a squared minus b squared,"},{"Start":"10:27.905 ","End":"10:29.945","Text":"and if this is a and this is b,"},{"Start":"10:29.945 ","End":"10:34.010","Text":"what we get as a squared is the square root of x squared,"},{"Start":"10:34.010 ","End":"10:38.970","Text":"which is just x, and 1 squared is 1."},{"Start":"10:38.970 ","End":"10:43.720","Text":"What we get in the denominator is just x minus 1."},{"Start":"10:43.720 ","End":"10:53.420","Text":"In the numerator, we get x minus 1 times square root of x plus 1, dx."},{"Start":"10:53.420 ","End":"10:58.275","Text":"This works out very well for us because we now have,"},{"Start":"10:58.275 ","End":"11:00.460","Text":"then x minus 1, if I put a bracket,"},{"Start":"11:00.460 ","End":"11:01.915","Text":"so you\u0027ll see it even better,"},{"Start":"11:01.915 ","End":"11:04.950","Text":"cancels with x minus 1,"},{"Start":"11:04.950 ","End":"11:07.950","Text":"and all we\u0027re left with is the square root of x plus 1."},{"Start":"11:07.950 ","End":"11:09.940","Text":"We\u0027ve turned a quotient,"},{"Start":"11:09.940 ","End":"11:12.235","Text":"this dividing line into a plus,"},{"Start":"11:12.235 ","End":"11:14.290","Text":"and this we know how to deal with."},{"Start":"11:14.290 ","End":"11:17.740","Text":"What we\u0027re left with basically is the"},{"Start":"11:17.740 ","End":"11:22.630","Text":"integral of and now using instead of the square root will use powers."},{"Start":"11:22.630 ","End":"11:28.025","Text":"We have x to the power of 1/2 plus 1, dx."},{"Start":"11:28.025 ","End":"11:29.670","Text":"Now because we have the sum,"},{"Start":"11:29.670 ","End":"11:34.210","Text":"we just have to take the integral of each of the 2 pieces separately and add them."},{"Start":"11:34.210 ","End":"11:36.820","Text":"This is x to the power of 1/2,"},{"Start":"11:36.820 ","End":"11:43.070","Text":"so its integral is x to the 0.5 plus 1 is 3 over 2,"},{"Start":"11:43.070 ","End":"11:45.005","Text":"over 3 over 2."},{"Start":"11:45.005 ","End":"11:48.335","Text":"The integral of a constant is that constant times x,"},{"Start":"11:48.335 ","End":"11:52.435","Text":"so it\u0027s just plus x plus c of course."},{"Start":"11:52.435 ","End":"11:58.620","Text":"Really this is the end of the integration those are the ones can simplify."},{"Start":"11:58.620 ","End":"12:02.255","Text":"Well, for example, instead of dividing by 3 over 2,"},{"Start":"12:02.255 ","End":"12:04.010","Text":"we could put 2/3,"},{"Start":"12:04.010 ","End":"12:06.109","Text":"and instead of an exponent,"},{"Start":"12:06.109 ","End":"12:09.500","Text":"we could put square root of x cubed, for example,"},{"Start":"12:09.500 ","End":"12:12.470","Text":"still plus x plus c. But this is really"},{"Start":"12:12.470 ","End":"12:16.250","Text":"the answer to the integration and the rest is simplification."},{"Start":"12:16.250 ","End":"12:17.750","Text":"We\u0027ve done part c,"},{"Start":"12:17.750 ","End":"12:20.700","Text":"and that\u0027s the end of this exercise."}],"ID":6700},{"Watched":false,"Name":"Exercise 10","Duration":"4m 16s","ChapterTopicVideoID":6642,"CourseChapterTopicPlaylistID":82571,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.com/Images/Videos_Thumbnails/6642.jpeg","UploadDate":"2016-07-19T11:50:04.4800000","DurationForVideoObject":"PT4M16S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:03.525","Text":"In this exercise, we have to compute the following integrals."},{"Start":"00:03.525 ","End":"00:04.620","Text":"There are 3 of them, a, b,"},{"Start":"00:04.620 ","End":"00:07.080","Text":"and c. Let\u0027s begin with a,"},{"Start":"00:07.080 ","End":"00:12.885","Text":"where we have the integral of 4 over x dx."},{"Start":"00:12.885 ","End":"00:16.620","Text":"Now as we know, a constant can come out in front of the integral."},{"Start":"00:16.620 ","End":"00:21.449","Text":"We just have 4 of the integral of 1 over x dx."},{"Start":"00:21.449 ","End":"00:23.220","Text":"This is an immediate integral."},{"Start":"00:23.220 ","End":"00:27.375","Text":"The integral of 1 over x is the natural log"},{"Start":"00:27.375 ","End":"00:32.460","Text":"of absolute value of x but we have to still put the 4 in."},{"Start":"00:32.460 ","End":"00:34.485","Text":"It\u0027s 4 and at the end,"},{"Start":"00:34.485 ","End":"00:38.625","Text":"we add a plus c. That\u0027s all there is to part a."},{"Start":"00:38.625 ","End":"00:41.415","Text":"In part b, and I\u0027ll copy it here,"},{"Start":"00:41.415 ","End":"00:47.560","Text":"we have the integral of 1 plus x over x squared dx."},{"Start":"00:47.560 ","End":"00:52.610","Text":"What we have to do is a little bit of algebra before we can use the standard formulas."},{"Start":"00:52.610 ","End":"00:57.680","Text":"We don\u0027t have a rule for f over g for some quotient rule for integrals."},{"Start":"00:57.680 ","End":"01:02.090","Text":"We\u0027ll just have to divide out and we\u0027ll get the integral"},{"Start":"01:02.090 ","End":"01:08.015","Text":"1 over x squared plus x over x squared dx."},{"Start":"01:08.015 ","End":"01:14.240","Text":"Then what we have here is 1 over x squared is x to the minus 2."},{"Start":"01:14.240 ","End":"01:18.050","Text":"I write it this way so we can use the formula for exponents."},{"Start":"01:18.050 ","End":"01:21.440","Text":"X over x squared is 1 over x,"},{"Start":"01:21.440 ","End":"01:26.790","Text":"which I\u0027ll leave as is because we know what to do with 1 over x, just like here."},{"Start":"01:27.490 ","End":"01:33.540","Text":"By the formula for x^n will be x to the,"},{"Start":"01:33.540 ","End":"01:35.025","Text":"minus 2 plus 1,"},{"Start":"01:35.025 ","End":"01:39.105","Text":"is minus 1 over minus 1."},{"Start":"01:39.105 ","End":"01:41.375","Text":"The 1 over x, as here,"},{"Start":"01:41.375 ","End":"01:46.790","Text":"gives us the natural log of absolute value of x and as usual,"},{"Start":"01:46.790 ","End":"01:49.685","Text":"plus c. That\u0027s basically it."},{"Start":"01:49.685 ","End":"01:51.650","Text":"If we want to simplify,"},{"Start":"01:51.650 ","End":"01:55.100","Text":"we could simplify this expression,"},{"Start":"01:55.100 ","End":"01:57.380","Text":"this bit here, I\u0027ll just write it at the side,"},{"Start":"01:57.380 ","End":"02:05.750","Text":"would come out as minus 1 over x instead of x to the minus 1 over minus 1."},{"Start":"02:05.750 ","End":"02:11.330","Text":"But that\u0027s optional in the simplification and I would leave this as the answer."},{"Start":"02:11.330 ","End":"02:16.475","Text":"Finally, we have part c. I\u0027ll just show you that I didn\u0027t cheat."},{"Start":"02:16.475 ","End":"02:19.885","Text":"There it is 1 plus 1 over x squared."},{"Start":"02:19.885 ","End":"02:23.655","Text":"That\u0027s down here and that\u0027s the last 1 of the 3."},{"Start":"02:23.655 ","End":"02:26.420","Text":"Once again, there is no shortcut."},{"Start":"02:26.420 ","End":"02:31.530","Text":"There is no formula for the integral of something squared."},{"Start":"02:31.530 ","End":"02:33.359","Text":"That doesn\u0027t exist."},{"Start":"02:33.359 ","End":"02:38.000","Text":"What we\u0027re going to have to do is a bit of algebra and expand the brackets."},{"Start":"02:38.000 ","End":"02:47.265","Text":"I\u0027ll just remind you that a plus b squared is a squared plus 2ab plus b squared."},{"Start":"02:47.265 ","End":"02:48.875","Text":"If we apply that here,"},{"Start":"02:48.875 ","End":"02:52.670","Text":"we get that this is the integral of 1 squared is"},{"Start":"02:52.670 ","End":"02:58.745","Text":"1 plus twice 1 times 1 over x is twice 1 over x."},{"Start":"02:58.745 ","End":"03:05.640","Text":"Then 1 over x squared is just 1 over x squared, all this, dx."},{"Start":"03:05.640 ","End":"03:10.070","Text":"One little bit of simplification before we do the integral of the first 2 terms,"},{"Start":"03:10.070 ","End":"03:11.990","Text":"I\u0027ll leave just as they are,"},{"Start":"03:11.990 ","End":"03:17.830","Text":"but I\u0027ll write this as x to the minus 2 so I can use the rule with the exponents."},{"Start":"03:17.830 ","End":"03:22.190","Text":"This is the simplification now the step of the integration."},{"Start":"03:22.190 ","End":"03:25.640","Text":"The integral of a constant is a constant times x,"},{"Start":"03:25.640 ","End":"03:29.195","Text":"so it\u0027s 1 x, which is just x."},{"Start":"03:29.195 ","End":"03:33.020","Text":"The integral of 1 over x we just did in part a is"},{"Start":"03:33.020 ","End":"03:37.635","Text":"the natural log of absolute value of x but because there\u0027s a 2 here,"},{"Start":"03:37.635 ","End":"03:43.564","Text":"the 2 stays, so it\u0027s the natural log of absolute value of x."},{"Start":"03:43.564 ","End":"03:47.015","Text":"Finally, because of the x to the power of,"},{"Start":"03:47.015 ","End":"03:56.455","Text":"we use the formula and it\u0027s x^minus 1 over the minus 1 plus c of course."},{"Start":"03:56.455 ","End":"03:58.430","Text":"This is the end of the exercise,"},{"Start":"03:58.430 ","End":"04:01.790","Text":"but those who would like to simplify it a bit could write it as"},{"Start":"04:01.790 ","End":"04:05.945","Text":"x plus twice log absolute value of x"},{"Start":"04:05.945 ","End":"04:15.780","Text":"minus 1 over x plus c. That\u0027s the third part and we\u0027re done."}],"ID":6701},{"Watched":false,"Name":"Exercise 11","Duration":"6m 33s","ChapterTopicVideoID":6643,"CourseChapterTopicPlaylistID":82571,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.com/Images/Videos_Thumbnails/6643.jpeg","UploadDate":"2016-07-19T11:50:45.1530000","DurationForVideoObject":"PT6M33S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:03.630","Text":"In this exercise, we have to compute the following integrals."},{"Start":"00:03.630 ","End":"00:04.800","Text":"There are 3 of them a, b,"},{"Start":"00:04.800 ","End":"00:10.755","Text":"and c. We are going to be using the same basic formula in all 3 of them."},{"Start":"00:10.755 ","End":"00:12.840","Text":"Let me just write it out here."},{"Start":"00:12.840 ","End":"00:17.805","Text":"This is the formula, I\u0027ve just written it here for the moment to get to it later."},{"Start":"00:17.805 ","End":"00:19.800","Text":"I\u0027ll just copy out number a,"},{"Start":"00:19.800 ","End":"00:26.340","Text":"which is the integral of 1 over 2x minus 10 dx."},{"Start":"00:26.340 ","End":"00:29.310","Text":"When I see this and I see this formula,"},{"Start":"00:29.310 ","End":"00:30.960","Text":"it looks rather similar."},{"Start":"00:30.960 ","End":"00:34.005","Text":"The important part of this formula is the a,"},{"Start":"00:34.005 ","End":"00:35.530","Text":"the coefficient of the x,"},{"Start":"00:35.530 ","End":"00:40.890","Text":"and I\u0027ll just highlight that so we can see that that is 2."},{"Start":"00:40.890 ","End":"00:44.290","Text":"Then we can see that a is equal to 2,"},{"Start":"00:44.290 ","End":"00:49.730","Text":"so we just plug it in and we get that this equals 1 over a,"},{"Start":"00:49.730 ","End":"00:53.420","Text":"which is 1 over 2 times natural log"},{"Start":"00:53.420 ","End":"00:58.220","Text":"of whatever was in the denominator goes inside the absolute value,"},{"Start":"00:58.220 ","End":"01:02.550","Text":"which is 2x minus 10 plus c at the end."},{"Start":"01:02.550 ","End":"01:05.085","Text":"That\u0027s all there is to part a."},{"Start":"01:05.085 ","End":"01:08.010","Text":"Now let\u0027s go on with part b."},{"Start":"01:08.010 ","End":"01:11.820","Text":"Part b is integral of"},{"Start":"01:11.820 ","End":"01:19.180","Text":"2x plus 2 over 2x plus 1 dx."},{"Start":"01:19.180 ","End":"01:23.990","Text":"Here again, we\u0027re faced with this dividing this quotient."},{"Start":"01:23.990 ","End":"01:27.860","Text":"There is no formula for f over g. We somehow have"},{"Start":"01:27.860 ","End":"01:32.295","Text":"to convert the division into addition or subtraction."},{"Start":"01:32.295 ","End":"01:33.930","Text":"I\u0027m going to show you a trick,"},{"Start":"01:33.930 ","End":"01:35.870","Text":"you might have thought of it yourself,"},{"Start":"01:35.870 ","End":"01:38.240","Text":"perhaps not, but it\u0027s a standard trick."},{"Start":"01:38.240 ","End":"01:40.040","Text":"The trick is as follows;"},{"Start":"01:40.040 ","End":"01:45.039","Text":"to say that if in the numerator I had 2x plus 1 also,"},{"Start":"01:45.039 ","End":"01:47.070","Text":"that would be great, I could cancel."},{"Start":"01:47.070 ","End":"01:49.650","Text":"Let\u0027s use that as a starting point."},{"Start":"01:49.650 ","End":"01:52.240","Text":"Integral and the dividing sign,"},{"Start":"01:52.240 ","End":"01:53.765","Text":"I\u0027ll make it a bit longer."},{"Start":"01:53.765 ","End":"01:56.165","Text":"We\u0027ll take the 2x plus 1 here,"},{"Start":"01:56.165 ","End":"01:59.935","Text":"and then the numerator we\u0027ll also put 2x plus 1."},{"Start":"01:59.935 ","End":"02:02.000","Text":"I\u0027ll put it in brackets you\u0027ll see."},{"Start":"02:02.000 ","End":"02:06.220","Text":"Now of course, we can\u0027t just go ahead and change an exercise,"},{"Start":"02:06.220 ","End":"02:08.255","Text":"this is not the same exercise."},{"Start":"02:08.255 ","End":"02:10.895","Text":"Because it was 2x plus 2 here,"},{"Start":"02:10.895 ","End":"02:16.210","Text":"all I have to do to make it right is add an extra plus 1 here."},{"Start":"02:16.210 ","End":"02:19.860","Text":"Now everything is right, because this is 2x plus 2 as it was before."},{"Start":"02:19.860 ","End":"02:22.620","Text":"I think you can see how this helps me, but if not,"},{"Start":"02:22.620 ","End":"02:25.545","Text":"let\u0027s just proceed as what we get is,"},{"Start":"02:25.545 ","End":"02:29.910","Text":"if we divide that we have the sum of 2 things over the same denominator,"},{"Start":"02:29.910 ","End":"02:34.160","Text":"so we take this over this plus this over this and convert it to an addition."},{"Start":"02:34.160 ","End":"02:37.960","Text":"2x plus 1 over 2x plus 1 is just 1."},{"Start":"02:37.960 ","End":"02:39.860","Text":"I\u0027m not going to spell it out."},{"Start":"02:39.860 ","End":"02:43.460","Text":"The second part is 1 over 2x plus 1."},{"Start":"02:43.460 ","End":"02:47.919","Text":"But we still have to keep our integration sign and the dx,"},{"Start":"02:47.919 ","End":"02:51.305","Text":"so we better put our brackets here, dx."},{"Start":"02:51.305 ","End":"02:52.860","Text":"Now we have an addition."},{"Start":"02:52.860 ","End":"02:54.420","Text":"Additions we know how to deal with,"},{"Start":"02:54.420 ","End":"02:57.035","Text":"we take the integral of each piece separately."},{"Start":"02:57.035 ","End":"02:59.750","Text":"The first piece which 1 is a constant,"},{"Start":"02:59.750 ","End":"03:01.735","Text":"it\u0027s just a constant times x,"},{"Start":"03:01.735 ","End":"03:04.005","Text":"1 times x is x."},{"Start":"03:04.005 ","End":"03:08.925","Text":"Here, we again have the same formula as here."},{"Start":"03:08.925 ","End":"03:11.820","Text":"This plus is this plus,"},{"Start":"03:11.820 ","End":"03:14.255","Text":"and 1 over 2x plus 1,"},{"Start":"03:14.255 ","End":"03:18.440","Text":"we use this formula again with a happening to be 2 again."},{"Start":"03:18.440 ","End":"03:26.660","Text":"It\u0027s 1 over 2 times natural logarithm of absolute value of whatever was here, 2x plus 1,"},{"Start":"03:26.660 ","End":"03:28.420","Text":"so it\u0027s 2x plus 1,"},{"Start":"03:28.420 ","End":"03:31.080","Text":"and at the end, of course, the plus"},{"Start":"03:31.080 ","End":"03:36.350","Text":"c. That does part b for us and that\u0027s a good trick to learn."},{"Start":"03:36.350 ","End":"03:40.250","Text":"Next, we move on to part c. I think I\u0027m going to run out of space,"},{"Start":"03:40.250 ","End":"03:43.135","Text":"so I better do some cleaning up here."},{"Start":"03:43.135 ","End":"03:47.540","Text":"I\u0027ve copied the exercise c over here and gotten rid of"},{"Start":"03:47.540 ","End":"03:52.070","Text":"the extra stuff and I think we\u0027ll keep this formula because it could be useful."},{"Start":"03:52.070 ","End":"03:55.550","Text":"Here we see something we haven\u0027t really seen before."},{"Start":"03:55.550 ","End":"04:00.155","Text":"I wouldn\u0027t know how to go about doing this unless I think of some trick."},{"Start":"04:00.155 ","End":"04:05.225","Text":"Fortunately, there is a trick and you may or may not have thought about it yourselves,"},{"Start":"04:05.225 ","End":"04:07.835","Text":"but if you look at this denominator,"},{"Start":"04:07.835 ","End":"04:10.130","Text":"this actually is a difference of squares."},{"Start":"04:10.130 ","End":"04:17.060","Text":"I can write 16x squared as something to the power of 2 and so I can write 1 as 1 squared."},{"Start":"04:17.060 ","End":"04:23.240","Text":"This reminds me of conjugates and difference of squares formula but in reverse."},{"Start":"04:23.240 ","End":"04:27.350","Text":"What I\u0027m saying is that if I have, for example,"},{"Start":"04:27.350 ","End":"04:31.295","Text":"a squared minus b squared,"},{"Start":"04:31.295 ","End":"04:32.810","Text":"I can write this formula,"},{"Start":"04:32.810 ","End":"04:37.440","Text":"which usually goes from right to left or left to right as a way of factoring."},{"Start":"04:37.440 ","End":"04:40.279","Text":"I can factor it as something times it\u0027s conjugate;"},{"Start":"04:40.279 ","End":"04:43.535","Text":"a plus b, a minus b,"},{"Start":"04:43.535 ","End":"04:45.305","Text":"or the other way around."},{"Start":"04:45.305 ","End":"04:49.610","Text":"All I have to do here is identify what\u0027s a and what\u0027s b."},{"Start":"04:49.610 ","End":"04:56.595","Text":"Now if I take in my case that a is what squared gives me 16x squared,"},{"Start":"04:56.595 ","End":"04:58.350","Text":"the answer will be 4x,"},{"Start":"04:58.350 ","End":"05:00.840","Text":"because if I take 4x and square it,"},{"Start":"05:00.840 ","End":"05:02.470","Text":"then I\u0027ve got 16x squared."},{"Start":"05:02.470 ","End":"05:05.670","Text":"If I take b as equal to 1,"},{"Start":"05:05.670 ","End":"05:08.445","Text":"then b squared is indeed 1."},{"Start":"05:08.445 ","End":"05:12.050","Text":"What I have here really is the a squared minus b"},{"Start":"05:12.050 ","End":"05:17.285","Text":"squared part and so I can now write it as the a plus ba minus b."},{"Start":"05:17.285 ","End":"05:21.920","Text":"This is the integral and I\u0027m leaving the dx as it is,"},{"Start":"05:21.920 ","End":"05:24.110","Text":"the integral sign as it is, the numerator,"},{"Start":"05:24.110 ","End":"05:25.370","Text":"I\u0027m leaving as it is,"},{"Start":"05:25.370 ","End":"05:28.850","Text":"but the denominator I\u0027m splitting according to this."},{"Start":"05:28.850 ","End":"05:30.835","Text":"I need a plus b,"},{"Start":"05:30.835 ","End":"05:34.935","Text":"and a plus b will be 4x plus 1."},{"Start":"05:34.935 ","End":"05:36.975","Text":"I need an a minus b,"},{"Start":"05:36.975 ","End":"05:40.005","Text":"which is 4x minus 1."},{"Start":"05:40.005 ","End":"05:42.575","Text":"Now I think it\u0027s clear what we have to do."},{"Start":"05:42.575 ","End":"05:46.810","Text":"We just have to cancel this with this."},{"Start":"05:46.810 ","End":"05:54.975","Text":"This is equal to the integral of 1 over 4x minus 1 dx."},{"Start":"05:54.975 ","End":"05:58.370","Text":"I\u0027ve highlighted the 4 in yellow to show you that we\u0027re going to"},{"Start":"05:58.370 ","End":"06:02.005","Text":"be using this formula this time with a as 4,"},{"Start":"06:02.005 ","End":"06:04.560","Text":"which makes a change, it was 2 before."},{"Start":"06:04.560 ","End":"06:06.950","Text":"Just looking at this formula,"},{"Start":"06:06.950 ","End":"06:11.895","Text":"we have to write down that this is equal to 1 over a,"},{"Start":"06:11.895 ","End":"06:20.595","Text":"which is 1 over 4, enough with the highlighting, think of it as yellow, times natural log."},{"Start":"06:20.595 ","End":"06:23.120","Text":"Then the absolute value I put whatever was here,"},{"Start":"06:23.120 ","End":"06:24.940","Text":"the 4x minus 1."},{"Start":"06:24.940 ","End":"06:28.805","Text":"At the end of course I add a plus c. That\u0027s it."},{"Start":"06:28.805 ","End":"06:30.950","Text":"That\u0027s the end of part c,"},{"Start":"06:30.950 ","End":"06:33.780","Text":"which is the end of this series."}],"ID":6702},{"Watched":false,"Name":"Exercise 12","Duration":"12m 23s","ChapterTopicVideoID":6644,"CourseChapterTopicPlaylistID":82571,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.com/Images/Videos_Thumbnails/6644.jpeg","UploadDate":"2016-07-19T11:52:04.0000000","DurationForVideoObject":"PT12M23S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:03.390","Text":"Here we have to compute the following integrals."},{"Start":"00:03.390 ","End":"00:07.455","Text":"3 of them, a, b, and c. I\u0027ve already written the formula."},{"Start":"00:07.455 ","End":"00:09.524","Text":"The main formula we\u0027re going to use,"},{"Start":"00:09.524 ","End":"00:11.940","Text":"which is this 1, you\u0027ve seen it before."},{"Start":"00:11.940 ","End":"00:14.490","Text":"Often a happens to be 1."},{"Start":"00:14.490 ","End":"00:18.255","Text":"I\u0027ve written the shortcut formula when a equals 1,"},{"Start":"00:18.255 ","End":"00:20.430","Text":"and we\u0027ll start with part a,"},{"Start":"00:20.430 ","End":"00:27.540","Text":"which is the integral of 4x plus 1 over x plus 2dx."},{"Start":"00:27.540 ","End":"00:32.250","Text":"Now since there\u0027s no rule for quotients for dividing by,"},{"Start":"00:32.250 ","End":"00:37.290","Text":"we have to find some way to convert this division into addition or subtraction."},{"Start":"00:37.290 ","End":"00:40.830","Text":"We have seen a trick used before,"},{"Start":"00:40.830 ","End":"00:43.235","Text":"and we\u0027re going to use it again here,"},{"Start":"00:43.235 ","End":"00:45.770","Text":"but here it\u0027s slightly more involved."},{"Start":"00:45.770 ","End":"00:50.285","Text":"Previously we had the same x plus something over x plus something."},{"Start":"00:50.285 ","End":"00:52.715","Text":"Here there\u0027s a 4x and here there\u0027s an x."},{"Start":"00:52.715 ","End":"00:54.725","Text":"Here\u0027s the way the trick works."},{"Start":"00:54.725 ","End":"00:57.950","Text":"The first thing we do is we take the 4"},{"Start":"00:57.950 ","End":"01:01.955","Text":"outside the brackets so that we have the same coefficient of x,"},{"Start":"01:01.955 ","End":"01:04.700","Text":"preferably just x on its own."},{"Start":"01:04.700 ","End":"01:07.490","Text":"What I\u0027m saying is I take the 4 out and I\u0027ll take the 4"},{"Start":"01:07.490 ","End":"01:10.794","Text":"completely outside the integral sign,"},{"Start":"01:10.794 ","End":"01:14.015","Text":"and then the denominator stays what it is,"},{"Start":"01:14.015 ","End":"01:16.685","Text":"x plus 2 and there\u0027s a dx."},{"Start":"01:16.685 ","End":"01:18.890","Text":"But if we take 4 out of here,"},{"Start":"01:18.890 ","End":"01:23.940","Text":"what we\u0027re left with is x plus 1/4."},{"Start":"01:23.940 ","End":"01:28.540","Text":"Now, what I would like for there to be x plus 2 here."},{"Start":"01:28.540 ","End":"01:30.955","Text":"What I\u0027m going to do,"},{"Start":"01:30.955 ","End":"01:33.190","Text":"and this is a trick you may have seen before,"},{"Start":"01:33.190 ","End":"01:40.225","Text":"is to keep this 4 and then write this as the integral of x plus."},{"Start":"01:40.225 ","End":"01:42.940","Text":"I\u0027m going to write what I want to be up here."},{"Start":"01:42.940 ","End":"01:44.185","Text":"What I wish I had."},{"Start":"01:44.185 ","End":"01:46.360","Text":"I wish I had x plus 2,"},{"Start":"01:46.360 ","End":"01:49.810","Text":"but I don\u0027t, I have x plus a 1/4."},{"Start":"01:49.810 ","End":"01:53.215","Text":"Or if we\u0027re going with decimals then plus 0.25."},{"Start":"01:53.215 ","End":"01:55.750","Text":"What I want is x plus 2 here."},{"Start":"01:55.750 ","End":"02:03.415","Text":"I can\u0027t just change the exercise but I\u0027d be okay if I compensate and subtract 1.75,"},{"Start":"02:03.415 ","End":"02:06.150","Text":"then this is 0.25 and all is well."},{"Start":"02:06.150 ","End":"02:08.570","Text":"The denominator I haven\u0027t touched."},{"Start":"02:08.570 ","End":"02:14.525","Text":"Now, what I\u0027m going to do is to split this fraction into 2 bits,"},{"Start":"02:14.525 ","End":"02:16.490","Text":"the bit with the x plus 2,"},{"Start":"02:16.490 ","End":"02:18.305","Text":"and I\u0027ll emphasize that here,"},{"Start":"02:18.305 ","End":"02:20.720","Text":"and the bit with the 1.75."},{"Start":"02:20.720 ","End":"02:23.374","Text":"When I have a difference over something,"},{"Start":"02:23.374 ","End":"02:25.025","Text":"I can take this over this,"},{"Start":"02:25.025 ","End":"02:26.765","Text":"minus this over this."},{"Start":"02:26.765 ","End":"02:28.480","Text":"Let\u0027s continue."},{"Start":"02:28.480 ","End":"02:31.385","Text":"What we get is 4,"},{"Start":"02:31.385 ","End":"02:33.635","Text":"and I\u0027ll take this over this."},{"Start":"02:33.635 ","End":"02:38.149","Text":"Now, x plus 2 over x plus 2 is just 1,"},{"Start":"02:38.149 ","End":"02:47.975","Text":"minus this over this is 1.75 over x plus 2 and all this dx."},{"Start":"02:47.975 ","End":"02:51.680","Text":"Now the thing about integrals is that now that we have a difference instead of"},{"Start":"02:51.680 ","End":"02:55.895","Text":"a quotient is that we can take the integral of each bit separately."},{"Start":"02:55.895 ","End":"03:00.590","Text":"We can take the integral of 1 and we can leave the 4 in front of that,"},{"Start":"03:00.590 ","End":"03:06.125","Text":"the integral of 1dx minus 4 times,"},{"Start":"03:06.125 ","End":"03:10.070","Text":"we can actually take this constant outside the integral."},{"Start":"03:10.070 ","End":"03:19.740","Text":"4 times 1.75 times the integral of 1 over x plus 2dx."},{"Start":"03:20.120 ","End":"03:26.600","Text":"Continuing. 1 is the integral of a constant like 1 is just 1x."},{"Start":"03:26.600 ","End":"03:29.090","Text":"It\u0027s 4 times 1 times x,"},{"Start":"03:29.090 ","End":"03:33.545","Text":"which is just 4x for the first bit and for the next bit,"},{"Start":"03:33.545 ","End":"03:37.080","Text":"multiplying 4 times 1.75,"},{"Start":"03:37.080 ","End":"03:40.785","Text":"that gives me 7, just simple computation."},{"Start":"03:40.785 ","End":"03:42.820","Text":"The 1 over x plus 2,"},{"Start":"03:42.820 ","End":"03:45.095","Text":"I\u0027m looking at this formula here,"},{"Start":"03:45.095 ","End":"03:51.500","Text":"gives me the natural log of the absolute value of this same x plus 2."},{"Start":"03:51.500 ","End":"03:53.810","Text":"At the end as always,"},{"Start":"03:53.810 ","End":"03:58.610","Text":"plus c. The next 1 is b,"},{"Start":"03:58.610 ","End":"04:01.685","Text":"and actually looks very similar to a."},{"Start":"04:01.685 ","End":"04:03.355","Text":"Let\u0027s clear up a bit."},{"Start":"04:03.355 ","End":"04:06.800","Text":"Now this 1 in b is somewhat similar to a,"},{"Start":"04:06.800 ","End":"04:10.820","Text":"but there\u0027s a difference in that here we\u0027ll have to take a constant 4"},{"Start":"04:10.820 ","End":"04:15.145","Text":"out of the numerator and a different constant 3 out of the denominator."},{"Start":"04:15.145 ","End":"04:17.465","Text":"That means we\u0027ll take 4 thirds out,"},{"Start":"04:17.465 ","End":"04:20.375","Text":"which will go in front of the integral sign."},{"Start":"04:20.375 ","End":"04:24.765","Text":"I\u0027ll start off by writing 4/3 integral of."},{"Start":"04:24.765 ","End":"04:27.075","Text":"Now the 4 I take out of here,"},{"Start":"04:27.075 ","End":"04:29.100","Text":"so what I\u0027m left with is,"},{"Start":"04:29.100 ","End":"04:32.355","Text":"when I take the 4 out of the 4x, I\u0027m left with x."},{"Start":"04:32.355 ","End":"04:35.190","Text":"Here I\u0027m left with a quarter."},{"Start":"04:35.190 ","End":"04:38.475","Text":"On the denominator, I\u0027ve taken 3 out."},{"Start":"04:38.475 ","End":"04:40.000","Text":"After I\u0027ve taken that,"},{"Start":"04:40.000 ","End":"04:41.530","Text":"I\u0027m left with x here,"},{"Start":"04:41.530 ","End":"04:45.190","Text":"but here I\u0027m left with 2 /3 dx, of course."},{"Start":"04:45.190 ","End":"04:52.765","Text":"Now, what I wish was that I had in the numerator also x plus 2/3, but I don\u0027t."},{"Start":"04:52.765 ","End":"04:54.610","Text":"As before with the trick,"},{"Start":"04:54.610 ","End":"04:58.375","Text":"what we do is we put what we would like there."},{"Start":"04:58.375 ","End":"05:02.320","Text":"That would be x plus 2/3,"},{"Start":"05:02.320 ","End":"05:04.390","Text":"and then we\u0027ll soon fix it."},{"Start":"05:04.390 ","End":"05:08.365","Text":"In the denominator, we\u0027ll have as before,"},{"Start":"05:08.365 ","End":"05:12.220","Text":"x plus 2/3 dx."},{"Start":"05:12.220 ","End":"05:16.045","Text":"What I\u0027m going to have to do is subtract something here."},{"Start":"05:16.045 ","End":"05:18.825","Text":"That it stays 1/4."},{"Start":"05:18.825 ","End":"05:21.140","Text":"Now I have an exercise in fractions."},{"Start":"05:21.140 ","End":"05:25.790","Text":"At the side, if I do what is 2/3 minus a 1/4,"},{"Start":"05:25.790 ","End":"05:27.425","Text":"back to fraction days,"},{"Start":"05:27.425 ","End":"05:31.670","Text":"I won\u0027t waste your time with the computation I\u0027ll just give you the answer,"},{"Start":"05:31.670 ","End":"05:34.760","Text":"which actually is 5/12."},{"Start":"05:34.760 ","End":"05:41.295","Text":"At least my way of thinking is that 2/3 is 8/12 and 1/4 is 3/12,"},{"Start":"05:41.295 ","End":"05:43.875","Text":"so 8 minus 3 is 5."},{"Start":"05:43.875 ","End":"05:47.335","Text":"If I put here minus 5/12,"},{"Start":"05:47.335 ","End":"05:50.690","Text":"all will be well and everything will be the same."},{"Start":"05:50.690 ","End":"05:54.030","Text":"The advantage is that now I have a difference."},{"Start":"05:54.030 ","End":"05:56.030","Text":"I don\u0027t have to deal with quotients,"},{"Start":"05:56.030 ","End":"05:57.260","Text":"I have a difference."},{"Start":"05:57.260 ","End":"06:02.105","Text":"Now I can split this up along the difference as follows."},{"Start":"06:02.105 ","End":"06:05.615","Text":"I\u0027ll leave the 4/3 outside the brackets."},{"Start":"06:05.615 ","End":"06:08.480","Text":"Now I have the integral of this over this,"},{"Start":"06:08.480 ","End":"06:14.810","Text":"this over is 1dx less the integral of the second bit."},{"Start":"06:14.810 ","End":"06:18.080","Text":"Because the integral of the difference is the difference of the integrals."},{"Start":"06:18.080 ","End":"06:26.100","Text":"I have now the integral of 5/12 over x plus 2/3 the dx,"},{"Start":"06:26.100 ","End":"06:28.275","Text":"of course, I\u0027ll stick it here."},{"Start":"06:28.275 ","End":"06:33.514","Text":"Now what we\u0027ll do is I\u0027ll actually separate it into 2 pieces."},{"Start":"06:33.514 ","End":"06:39.720","Text":"The first piece is 4/3 times the integral of 1dx."},{"Start":"06:39.720 ","End":"06:42.960","Text":"The second piece is,"},{"Start":"06:42.960 ","End":"06:49.935","Text":"I can bring the 5/12 out in front and write it as 5/12 times 4/3."},{"Start":"06:49.935 ","End":"06:52.820","Text":"Let\u0027s see, 4 times 5 is 20,"},{"Start":"06:52.820 ","End":"06:56.930","Text":"and 3 times 12 is 36."},{"Start":"06:56.930 ","End":"07:06.350","Text":"20 over 36 times the integral of 1 over x plus 2/3 dx."},{"Start":"07:06.350 ","End":"07:10.975","Text":"All I did was multiplied each of these pieces by 4/3,"},{"Start":"07:10.975 ","End":"07:15.420","Text":"and I also took the 5/12 out of the integration sign."},{"Start":"07:15.420 ","End":"07:19.960","Text":"Then I did at the side 4/3 times 5/12."},{"Start":"07:19.960 ","End":"07:23.630","Text":"Another fraction exercise is 20 over 36."},{"Start":"07:23.630 ","End":"07:27.290","Text":"Actually, I could divide top and bottom by 4 if I\u0027m"},{"Start":"07:27.290 ","End":"07:32.090","Text":"to this sort of thing and I could get 5 over 9."},{"Start":"07:32.090 ","End":"07:35.570","Text":"I see I accidentally erased the formula I need."},{"Start":"07:35.570 ","End":"07:37.805","Text":"I\u0027ll just put it back in and erase the clutter."},{"Start":"07:37.805 ","End":"07:42.785","Text":"Hang on. Yeah, this is the formula I accidentally erased. Here it is again."},{"Start":"07:42.785 ","End":"07:46.135","Text":"Now let\u0027s get onto the actual integration."},{"Start":"07:46.135 ","End":"07:50.705","Text":"The integral of a constant is just the constant times x,"},{"Start":"07:50.705 ","End":"07:53.645","Text":"so just 1x or x."},{"Start":"07:53.645 ","End":"07:55.355","Text":"But there\u0027s a 4/3 there,"},{"Start":"07:55.355 ","End":"07:59.240","Text":"so it\u0027s 4/3 x for this part."},{"Start":"07:59.240 ","End":"08:01.610","Text":"Now, for the second part,"},{"Start":"08:01.610 ","End":"08:03.860","Text":"what we have is minus,"},{"Start":"08:03.860 ","End":"08:05.480","Text":"just feel like simplifying it,"},{"Start":"08:05.480 ","End":"08:08.570","Text":"divide the top and bottom by 4, 5/9."},{"Start":"08:08.570 ","End":"08:11.375","Text":"Here\u0027s the part where we need the formula."},{"Start":"08:11.375 ","End":"08:13.970","Text":"This is like the 1 over x plus b."},{"Start":"08:13.970 ","End":"08:20.555","Text":"We have natural logarithm of the same thing that was here, x plus 2/3."},{"Start":"08:20.555 ","End":"08:22.085","Text":"Finally at the end,"},{"Start":"08:22.085 ","End":"08:27.830","Text":"we put plus c, so that is part b out of 3."},{"Start":"08:27.830 ","End":"08:30.160","Text":"Now let\u0027s get onto the third part."},{"Start":"08:30.160 ","End":"08:33.315","Text":"Let\u0027s see what is the third part."},{"Start":"08:33.315 ","End":"08:35.585","Text":"Here we are, copied this here,"},{"Start":"08:35.585 ","End":"08:36.770","Text":"got rid of the extra junk,"},{"Start":"08:36.770 ","End":"08:40.310","Text":"and I brought back the formula from before that I\u0027m going to need."},{"Start":"08:40.310 ","End":"08:42.215","Text":"Let\u0027s see what can we do here."},{"Start":"08:42.215 ","End":"08:44.960","Text":"Again, we\u0027re plagued with this quotient,"},{"Start":"08:44.960 ","End":"08:46.640","Text":"something divided by something,"},{"Start":"08:46.640 ","End":"08:49.670","Text":"and there\u0027s no integration rule for something over something."},{"Start":"08:49.670 ","End":"08:55.035","Text":"So we have to somehow convert the quotient to a sum or difference."},{"Start":"08:55.035 ","End":"08:59.240","Text":"As usual, there\u0027s a trick that once you\u0027ve seen the trick once,"},{"Start":"08:59.240 ","End":"09:00.830","Text":"you can use it several times."},{"Start":"09:00.830 ","End":"09:04.370","Text":"Here, the trick is to factorize the denominator,"},{"Start":"09:04.370 ","End":"09:08.360","Text":"we have something linear here and here we have something quadratic."},{"Start":"09:08.360 ","End":"09:10.085","Text":"If we factorize it,"},{"Start":"09:10.085 ","End":"09:14.715","Text":"and if we get lucky enough that somehow we get something to cancel,"},{"Start":"09:14.715 ","End":"09:16.325","Text":"then we\u0027ll be in good shape."},{"Start":"09:16.325 ","End":"09:22.145","Text":"Now, let me remind you of how we factorize a quadratic expression."},{"Start":"09:22.145 ","End":"09:28.040","Text":"If we have ax squared plus bx plus c,"},{"Start":"09:28.040 ","End":"09:32.164","Text":"a quadratic polynomial, we can factorize it as follows."},{"Start":"09:32.164 ","End":"09:40.955","Text":"This is equal to a times x minus x1 times x minus x2."},{"Start":"09:40.955 ","End":"09:44.005","Text":"All I have to do is tell you what are x1 and x2."},{"Start":"09:44.005 ","End":"09:46.715","Text":"Well, those are the solutions of the equation."},{"Start":"09:46.715 ","End":"09:52.385","Text":"What I do is I take the equation ax squared plus bx plus c equals 0,"},{"Start":"09:52.385 ","End":"09:56.915","Text":"and then usually when you use the formula or whatever other method you use,"},{"Start":"09:56.915 ","End":"09:58.880","Text":"you get 2 solutions for x."},{"Start":"09:58.880 ","End":"10:01.745","Text":"You get x equals something or x equals something."},{"Start":"10:01.745 ","End":"10:05.900","Text":"X1 and x2 are the 2 solutions,"},{"Start":"10:05.900 ","End":"10:08.645","Text":"basically the solutions of this equation."},{"Start":"10:08.645 ","End":"10:10.640","Text":"Now, in our case,"},{"Start":"10:10.640 ","End":"10:14.005","Text":"what we get if we solve the equation,"},{"Start":"10:14.005 ","End":"10:19.940","Text":"2x squared plus x minus 3 equals 0,"},{"Start":"10:19.940 ","End":"10:24.770","Text":"then we get the 2 solutions happen to be x equals 1."},{"Start":"10:24.770 ","End":"10:26.859","Text":"Let\u0027s call that the x1,"},{"Start":"10:26.859 ","End":"10:30.500","Text":"and the other 1 happens to be minus 3."},{"Start":"10:30.500 ","End":"10:33.995","Text":"I\u0027m not going to solve for the quadratic equation for you,."},{"Start":"10:33.995 ","End":"10:35.420","Text":"I\u0027m assuming you can do that."},{"Start":"10:35.420 ","End":"10:39.525","Text":"The 2 solutions are 1 and minus 3."},{"Start":"10:39.525 ","End":"10:44.660","Text":"This means that this can now be factorized and I\u0027ll do it over here."},{"Start":"10:44.660 ","End":"10:46.070","Text":"But first, excuse me,"},{"Start":"10:46.070 ","End":"10:47.210","Text":"I made a small mistake."},{"Start":"10:47.210 ","End":"10:49.865","Text":"This is actually supposed to be minus 3 over 2."},{"Start":"10:49.865 ","End":"10:51.710","Text":"Anyway, getting back to here,"},{"Start":"10:51.710 ","End":"10:57.110","Text":"we get the integral of x minus 1 over."},{"Start":"10:57.110 ","End":"10:59.060","Text":"According to this formula,"},{"Start":"10:59.060 ","End":"11:04.675","Text":"I put a which is the 2 times x minus x1,"},{"Start":"11:04.675 ","End":"11:07.150","Text":"which is x minus 1,"},{"Start":"11:07.150 ","End":"11:09.890","Text":"times x minus x2,"},{"Start":"11:09.890 ","End":"11:13.890","Text":"which is x minus 3 over 2dx."},{"Start":"11:14.480 ","End":"11:19.595","Text":"Very good. Because what I can do is cancel"},{"Start":"11:19.595 ","End":"11:26.525","Text":"this x minus 1 with this x minus 1 and all I\u0027m left with here is a 1."},{"Start":"11:26.525 ","End":"11:29.509","Text":"What\u0027s more? I can do a little bit extra."},{"Start":"11:29.509 ","End":"11:34.760","Text":"I can actually multiply the 2 by the x minus 3 over 2."},{"Start":"11:34.760 ","End":"11:41.790","Text":"What I\u0027ll get is the integral of 1 over 2 times x is 2x,"},{"Start":"11:41.790 ","End":"11:44.805","Text":"2 times 3 over 2 is 3."},{"Start":"11:44.805 ","End":"11:48.675","Text":"I have 1 over 2x minus 3 dx."},{"Start":"11:48.675 ","End":"11:53.290","Text":"Now I can finally do the integration because I have this formula here."},{"Start":"11:53.290 ","End":"11:57.335","Text":"The a part which is the most important is 2."},{"Start":"11:57.335 ","End":"12:02.164","Text":"What I can say is that this equals 1 over 2,"},{"Start":"12:02.164 ","End":"12:03.975","Text":"because the 2 is the a,"},{"Start":"12:03.975 ","End":"12:08.240","Text":"natural log, and I just put this term in absolute value,"},{"Start":"12:08.240 ","End":"12:12.260","Text":"2x minus 3 and add a constant."},{"Start":"12:12.260 ","End":"12:14.180","Text":"This is the answer."},{"Start":"12:14.180 ","End":"12:16.595","Text":"This is a good trick to learn."},{"Start":"12:16.595 ","End":"12:19.270","Text":"Factorize if you have a quadratic here."},{"Start":"12:19.270 ","End":"12:24.030","Text":"That\u0027s done with part c and we\u0027re done with this exercise."}],"ID":6703},{"Watched":false,"Name":"Exercise 13","Duration":"6m 7s","ChapterTopicVideoID":6645,"CourseChapterTopicPlaylistID":82571,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.com/Images/Videos_Thumbnails/6645.jpeg","UploadDate":"2016-07-19T11:52:43.9530000","DurationForVideoObject":"PT6M7S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:03.900","Text":"In this exercise, we have to compute the following integrals."},{"Start":"00:03.900 ","End":"00:06.870","Text":"3 of them, a, b, and c. Then all of them will be"},{"Start":"00:06.870 ","End":"00:10.365","Text":"using the same basic formula, which is this."},{"Start":"00:10.365 ","End":"00:13.965","Text":"The integral of e to the power of ax plus b"},{"Start":"00:13.965 ","End":"00:18.060","Text":"is just 1 over a of the same thing plus the constant."},{"Start":"00:18.060 ","End":"00:22.860","Text":"Frequently b is missing and in this case I just wrote it as a separate rule."},{"Start":"00:22.860 ","End":"00:31.005","Text":"The integral of e^ax dx is just 1 over a of the same thing plus the constant at the end."},{"Start":"00:31.005 ","End":"00:33.560","Text":"In the first exercise,"},{"Start":"00:33.560 ","End":"00:39.380","Text":"the integral of e to the power of 4x plus 1 dx,"},{"Start":"00:39.380 ","End":"00:41.315","Text":"I\u0027m going to do a bit of coloring."},{"Start":"00:41.315 ","End":"00:45.470","Text":"The important thing is a in these formulas and in the exercise,"},{"Start":"00:45.470 ","End":"00:46.790","Text":"and that\u0027s what we divide by."},{"Start":"00:46.790 ","End":"00:51.800","Text":"We see in this example that a is 4 and so this is going to equal."},{"Start":"00:51.800 ","End":"00:55.040","Text":"We start with 1 over a, 1/4,"},{"Start":"00:55.040 ","End":"00:57.230","Text":"and then the same thing as here,"},{"Start":"00:57.230 ","End":"01:00.050","Text":"e to the power of 4x plus 1,"},{"Start":"01:00.050 ","End":"01:02.540","Text":"and finally plus a constant."},{"Start":"01:02.540 ","End":"01:04.870","Text":"That\u0027s it for part a."},{"Start":"01:04.870 ","End":"01:07.205","Text":"Now let\u0027s get on to part b,"},{"Start":"01:07.205 ","End":"01:09.800","Text":"the integral of sum of 3 things,"},{"Start":"01:09.800 ","End":"01:18.140","Text":"e to the x plus e to the 2x plus e to the 3x all over e^4x dx."},{"Start":"01:18.140 ","End":"01:19.760","Text":"Now what does this equal?"},{"Start":"01:19.760 ","End":"01:21.770","Text":"We should do a little bit of algebra first"},{"Start":"01:21.770 ","End":"01:24.170","Text":"because you can\u0027t use these rules straight away."},{"Start":"01:24.170 ","End":"01:27.630","Text":"We\u0027re going to use 2 rules. First, the distributive rule."},{"Start":"01:27.630 ","End":"01:30.630","Text":"This plus this plus this all over the same denominator,"},{"Start":"01:30.630 ","End":"01:33.885","Text":"so we divide each 1 over the e^4x."},{"Start":"01:33.885 ","End":"01:40.670","Text":"We get e^x over e^4x and if you use your rules of exponents,"},{"Start":"01:40.670 ","End":"01:43.190","Text":"x minus 4x is minus 3x."},{"Start":"01:43.190 ","End":"01:45.880","Text":"This is e to the minus 3x."},{"Start":"01:45.880 ","End":"01:48.780","Text":"When we divide, we subtract the exponents,"},{"Start":"01:48.780 ","End":"01:52.575","Text":"e^2x over e^4x is e to the 2 minus 4x,"},{"Start":"01:52.575 ","End":"01:55.410","Text":"which is e to the minus 2x."},{"Start":"01:55.410 ","End":"01:58.880","Text":"Then 3x minus 4x gives us minus x."},{"Start":"01:58.880 ","End":"02:02.420","Text":"So we have e to the minus x, which if I want,"},{"Start":"02:02.420 ","End":"02:08.805","Text":"I can write as e to the minus 1x and we still need the integral dx."},{"Start":"02:08.805 ","End":"02:12.455","Text":"This is good. We have the sum of 3 separate things,"},{"Start":"02:12.455 ","End":"02:14.300","Text":"and for each of them we have a formula."},{"Start":"02:14.300 ","End":"02:16.940","Text":"Let\u0027s just apply the integral to each 1"},{"Start":"02:16.940 ","End":"02:20.340","Text":"separately and then we put the common C at the end."},{"Start":"02:20.340 ","End":"02:24.000","Text":"For e^3x, we let a equal minus 3,"},{"Start":"02:24.000 ","End":"02:25.910","Text":"a we can use this 1 here."},{"Start":"02:25.910 ","End":"02:31.130","Text":"So we get 1 over minus 3 of the same thing,"},{"Start":"02:31.130 ","End":"02:33.175","Text":"e to the minus 3x,"},{"Start":"02:33.175 ","End":"02:39.045","Text":"and then 1 over minus 2 of e to the minus 2x,"},{"Start":"02:39.045 ","End":"02:42.025","Text":"and then 1 over minus 1,"},{"Start":"02:42.025 ","End":"02:44.945","Text":"I\u0027ll write it that way and simplify later,"},{"Start":"02:44.945 ","End":"02:50.705","Text":"of e to the minus 1x and then plus C at the end."},{"Start":"02:50.705 ","End":"02:53.495","Text":"Finally, if we want to simplify it,"},{"Start":"02:53.495 ","End":"02:55.870","Text":"we can put, for example,"},{"Start":"02:55.870 ","End":"02:57.740","Text":"this is the answer, but for example,"},{"Start":"02:57.740 ","End":"03:01.385","Text":"we could put the negative exponents on the denominator,"},{"Start":"03:01.385 ","End":"03:03.200","Text":"maybe get something like,"},{"Start":"03:03.200 ","End":"03:04.970","Text":"and put the minus out in front,"},{"Start":"03:04.970 ","End":"03:09.970","Text":"so minus 1 over 3 e to the 3x,"},{"Start":"03:09.970 ","End":"03:17.300","Text":"for example, minus 1 over 2 goes into the denominator and so it is e to the 2x."},{"Start":"03:17.300 ","End":"03:24.470","Text":"Then minus 1 over e to the x plus C. But this will do adequately."},{"Start":"03:24.470 ","End":"03:26.810","Text":"This is just a bit of simplification."},{"Start":"03:26.810 ","End":"03:29.990","Text":"Finally, we have the last 1,"},{"Start":"03:29.990 ","End":"03:33.930","Text":"and basically we\u0027re going to be using this formula."},{"Start":"03:33.930 ","End":"03:38.450","Text":"We just have to do some algebra to bring each of these exponents."},{"Start":"03:38.450 ","End":"03:41.195","Text":"We have square roots or cube roots,"},{"Start":"03:41.195 ","End":"03:44.030","Text":"and we have stuff in the denominator and using the rules of"},{"Start":"03:44.030 ","End":"03:47.155","Text":"exponents will get each of the pieces into this form."},{"Start":"03:47.155 ","End":"03:49.955","Text":"What we have here is first of all,"},{"Start":"03:49.955 ","End":"03:51.455","Text":"we have 4 times something."},{"Start":"03:51.455 ","End":"03:55.715","Text":"Now the 4 is a constant and can come out of the integration sign."},{"Start":"03:55.715 ","End":"03:57.560","Text":"Also because we have a plus,"},{"Start":"03:57.560 ","End":"04:00.740","Text":"we can take the integral of this separately and this separately."},{"Start":"04:00.740 ","End":"04:01.805","Text":"For the first bit,"},{"Start":"04:01.805 ","End":"04:08.045","Text":"the 4 comes out and the square root of e^x is e to the x to the power of 1/2,"},{"Start":"04:08.045 ","End":"04:10.370","Text":"I can just do this as a side exercise."},{"Start":"04:10.370 ","End":"04:11.975","Text":"E to the power of x,"},{"Start":"04:11.975 ","End":"04:14.495","Text":"the square root makes it to the power of 1/2."},{"Start":"04:14.495 ","End":"04:21.940","Text":"multiply the exponents, and we get e to the power of 1/2x."},{"Start":"04:21.940 ","End":"04:27.335","Text":"Here we have e to the power of 1/2x and dx."},{"Start":"04:27.335 ","End":"04:32.420","Text":"The second bit is the cube root of e to the 4x 1 over."},{"Start":"04:32.420 ","End":"04:37.735","Text":"1 over the cube root of e to the 4x,"},{"Start":"04:37.735 ","End":"04:44.460","Text":"what we get is 1 over e to the power of 4x over 3."},{"Start":"04:45.140 ","End":"04:47.835","Text":"The 1 over makes it negative,"},{"Start":"04:47.835 ","End":"04:51.845","Text":"so it\u0027s e to the power of minus 4x over 3,"},{"Start":"04:51.845 ","End":"04:54.125","Text":"which are immediately right in here."},{"Start":"04:54.125 ","End":"04:55.855","Text":"That\u0027s the second integral."},{"Start":"04:55.855 ","End":"04:57.285","Text":"So e to the,"},{"Start":"04:57.285 ","End":"05:03.015","Text":"and I\u0027ll write it as minus 4/3 times the x so it looks like this."},{"Start":"05:03.015 ","End":"05:05.810","Text":"We\u0027ll have twice the use of this rule."},{"Start":"05:05.810 ","End":"05:10.085","Text":"Once with a being 1/2 and once with a being minus 4/3,"},{"Start":"05:10.085 ","End":"05:12.905","Text":"I forgot the dx over here."},{"Start":"05:12.905 ","End":"05:16.920","Text":"The first 1, e to the power of 1/2x,"},{"Start":"05:16.920 ","End":"05:21.485","Text":"its integral is 1 over 1/2 and what is 1 over 1/2?"},{"Start":"05:21.485 ","End":"05:24.730","Text":"1 Over 1/2 is just 2."},{"Start":"05:24.730 ","End":"05:26.375","Text":"So here we have 2,"},{"Start":"05:26.375 ","End":"05:33.230","Text":"which combines with the 4 to give 8e to the power of 1/2x."},{"Start":"05:33.230 ","End":"05:36.395","Text":"The second bit is using this formula,"},{"Start":"05:36.395 ","End":"05:46.000","Text":"1 over minus 4/3 and 1 over minus 4/3 is minus 3/4."},{"Start":"05:46.000 ","End":"05:50.540","Text":"Instead of the plus, I\u0027ll erase it and write it as minus and then"},{"Start":"05:50.540 ","End":"05:58.385","Text":"the 3/4 e to the power of minus 4/3x and then finally,"},{"Start":"05:58.385 ","End":"06:02.330","Text":"we just write plus C. It\u0027s possible to get rid of"},{"Start":"06:02.330 ","End":"06:08.470","Text":"the negative exponents and write it in terms of roots but we could end it here."}],"ID":6704},{"Watched":false,"Name":"Exercise 14","Duration":"11m 9s","ChapterTopicVideoID":6646,"CourseChapterTopicPlaylistID":82571,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.com/Images/Videos_Thumbnails/6646.jpeg","UploadDate":"2017-02-14T01:55:36.4070000","DurationForVideoObject":"PT11M9S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:04.290","Text":"In this exercise, we have to compute the following integrals a, b,"},{"Start":"00:04.290 ","End":"00:09.660","Text":"and c, and I\u0027ve written down the formulas that we\u0027re going to need for this exercise."},{"Start":"00:09.660 ","End":"00:12.660","Text":"I have the feeling we\u0027ve done a before but never mind,"},{"Start":"00:12.660 ","End":"00:13.890","Text":"we\u0027ll do it again."},{"Start":"00:13.890 ","End":"00:17.190","Text":"You would think that there might be a rule for something squared,"},{"Start":"00:17.190 ","End":"00:18.960","Text":"some kind of template rule,"},{"Start":"00:18.960 ","End":"00:22.140","Text":"unfortunately, there is no such rule, and instead,"},{"Start":"00:22.140 ","End":"00:25.950","Text":"we\u0027re just going to use the laws of algebra which is to"},{"Start":"00:25.950 ","End":"00:31.720","Text":"say that a to the power of b to the power of c."}],"ID":6705},{"Watched":false,"Name":"Exercise 15","Duration":"6m 57s","ChapterTopicVideoID":1516,"CourseChapterTopicPlaylistID":82571,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.com/Images/Videos_Thumbnails/1516.jpeg","UploadDate":"2017-01-25T23:56:07.4870000","DurationForVideoObject":"PT6M57S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:04.590","Text":"In this exercise, we have to compute the following integrals, a, b, and c."},{"Start":"00:04.590 ","End":"00:07.200","Text":"I\u0027ve written part a over here"},{"Start":"00:07.200 ","End":"00:10.590","Text":"and I looked ahead to see which formula I\u0027m going to need,"},{"Start":"00:10.590 ","End":"00:12.270","Text":"and I wrote it over here,"},{"Start":"00:12.270 ","End":"00:14.010","Text":"and we shall soon see."},{"Start":"00:14.010 ","End":"00:18.735","Text":"What we have here is 1 over 1 plus 4x squared"},{"Start":"00:18.735 ","End":"00:24.480","Text":"and what I would like is 1 over 1 plus something squared,"},{"Start":"00:24.480 ","End":"00:27.540","Text":"because that reminds me of the arctangent rule."},{"Start":"00:27.540 ","End":"00:30.795","Text":"Let\u0027s just write this or just look at here,"},{"Start":"00:30.795 ","End":"00:31.860","Text":"at the rule that I wrote,"},{"Start":"00:31.860 ","End":"00:35.835","Text":"I want 1 over which I have, 1 plus,"},{"Start":"00:35.835 ","End":"00:40.350","Text":"which I have and what I need is something squared."},{"Start":"00:40.350 ","End":"00:42.025","Text":"In fact, this part,"},{"Start":"00:42.025 ","End":"00:46.380","Text":"instead of the 4x squared, is 2x squared."},{"Start":"00:46.380 ","End":"00:52.550","Text":"I\u0027ll write this as 2x, in brackets, squared and dx."},{"Start":"00:52.550 ","End":"00:56.390","Text":"Now this rule says that if I have instead of x here,"},{"Start":"00:56.390 ","End":"00:58.145","Text":"I have ax plus b,"},{"Start":"00:58.145 ","End":"01:01.420","Text":"then I just get the arctangent of what\u0027s here,"},{"Start":"01:01.420 ","End":"01:02.975","Text":"but, with the 1 over a."},{"Start":"01:02.975 ","End":"01:04.705","Text":"In fact, let me highlight that."},{"Start":"01:04.705 ","End":"01:07.545","Text":"This rule comes from the arctangent rule,"},{"Start":"01:07.545 ","End":"01:09.010","Text":"when it\u0027s just an x in here,"},{"Start":"01:09.010 ","End":"01:10.875","Text":"but if there\u0027s an ax plus b,"},{"Start":"01:10.875 ","End":"01:14.660","Text":"then we have to divide by this a here."},{"Start":"01:14.660 ","End":"01:16.280","Text":"This is what we have,"},{"Start":"01:16.280 ","End":"01:20.640","Text":"we have our a here is being equal to 2"},{"Start":"01:20.640 ","End":"01:25.184","Text":"and there is no b or b is 0, but that\u0027s less important."},{"Start":"01:25.184 ","End":"01:28.760","Text":"Now I can write that this thing is equal to,"},{"Start":"01:28.760 ","End":"01:31.625","Text":"according to this rule, to 1 over a,"},{"Start":"01:31.625 ","End":"01:38.030","Text":"which is 1/2 of the arctangent and the ax plus b is just this expression,"},{"Start":"01:38.030 ","End":"01:40.710","Text":"the 2x and plus c,"},{"Start":"01:40.710 ","End":"01:43.470","Text":"that\u0027s all there is to part a."},{"Start":"01:43.470 ","End":"01:50.220","Text":"Part b looks very similar to 1 over the square root of 1 minus x squared,"},{"Start":"01:50.220 ","End":"01:53.415","Text":"but there\u0027s a 4 and it\u0027s not a 1,"},{"Start":"01:53.415 ","End":"01:55.400","Text":"so we\u0027ll have to do a bit of algebra"},{"Start":"01:55.400 ","End":"02:01.215","Text":"and say that this equals integral of 1 over the square root,"},{"Start":"02:01.215 ","End":"02:03.810","Text":"because I want there to be a 1,"},{"Start":"02:03.810 ","End":"02:07.095","Text":"the obvious thing to do is to take 4 outside the brackets."},{"Start":"02:07.095 ","End":"02:16.485","Text":"I take 4 outside the brackets and what I\u0027m left with is 1 minus x squared over 4, dx."},{"Start":"02:16.485 ","End":"02:20.420","Text":"I get this slowly to look like this formula here,"},{"Start":"02:20.420 ","End":"02:24.170","Text":"maybe with some extras, 4 times something."},{"Start":"02:24.170 ","End":"02:28.700","Text":"It\u0027s 1 over square root of 4 times the square root of this,"},{"Start":"02:28.700 ","End":"02:31.070","Text":"so I can take the square root of 4 is 2"},{"Start":"02:31.070 ","End":"02:37.370","Text":"and then I have the square root of 1 minus x squared over 4"},{"Start":"02:37.370 ","End":"02:39.410","Text":"and the x squared over 4,"},{"Start":"02:39.410 ","End":"02:43.580","Text":"I can actually write as x over 2 squared,"},{"Start":"02:43.580 ","End":"02:48.800","Text":"because it\u0027s x squared over 2 squared dx, I\u0027m getting closer."},{"Start":"02:48.800 ","End":"02:51.950","Text":"Now this equals, 2 I can take outside,"},{"Start":"02:51.950 ","End":"02:55.090","Text":"or 1/2 I can take outside the integral,"},{"Start":"02:55.090 ","End":"03:01.550","Text":"so it\u0027s 1/2 times the integral of 1 over, I put the dx here,"},{"Start":"03:01.550 ","End":"03:04.645","Text":"what I want is something that looks like this."},{"Start":"03:04.645 ","End":"03:10.445","Text":"If I write it as a square root of 1 minus,"},{"Start":"03:10.445 ","End":"03:13.190","Text":"and let my a be a 1/2,"},{"Start":"03:13.190 ","End":"03:20.045","Text":"so I have a 1/2x and there is no b or it\u0027s plus 0 squared,"},{"Start":"03:20.045 ","End":"03:27.455","Text":"then I\u0027ll be able to use this formula because this 1/2 will be my a from here."},{"Start":"03:27.455 ","End":"03:35.505","Text":"Now I can apply the integral and get that this equals 1/2 times arc sine,"},{"Start":"03:35.505 ","End":"03:37.370","Text":"sorry, I need the 1 over a,"},{"Start":"03:37.370 ","End":"03:40.655","Text":"which is 1 over the 1/2,"},{"Start":"03:40.655 ","End":"03:43.150","Text":"and then the arc sine,"},{"Start":"03:43.150 ","End":"03:45.240","Text":"then the ax plus b,"},{"Start":"03:45.240 ","End":"03:48.675","Text":"which is whatever expression is in brackets the 1/2x,"},{"Start":"03:48.675 ","End":"03:51.210","Text":"and finally plus C."},{"Start":"03:51.210 ","End":"03:52.725","Text":"Clean it up a bit,"},{"Start":"03:52.725 ","End":"03:54.105","Text":"this doesn\u0027t look right."},{"Start":"03:54.105 ","End":"03:56.640","Text":"1/2 times 1 over 1/2 is just 1,"},{"Start":"03:56.640 ","End":"04:01.335","Text":"so all we\u0027re left with is arc sine of x over 2 plus c,"},{"Start":"04:01.335 ","End":"04:04.575","Text":"and that\u0027s the answer for part b."},{"Start":"04:04.575 ","End":"04:08.370","Text":"Here we are with part c,"},{"Start":"04:08.370 ","End":"04:11.030","Text":"and I started to solve this on my own"},{"Start":"04:11.030 ","End":"04:15.425","Text":"and looked ahead in time that this is the formula that we\u0027re going to need,"},{"Start":"04:15.425 ","End":"04:19.120","Text":"which is the integral of 1 over 1 minus x squared,"},{"Start":"04:19.120 ","End":"04:20.610","Text":"and never mind what it equals,"},{"Start":"04:20.610 ","End":"04:22.140","Text":"it\u0027s some mess here,"},{"Start":"04:22.140 ","End":"04:23.700","Text":"but it\u0027s not what we have,"},{"Start":"04:23.700 ","End":"04:26.085","Text":"we have x squared."},{"Start":"04:26.085 ","End":"04:28.995","Text":"I played around a bit and thought,"},{"Start":"04:28.995 ","End":"04:30.600","Text":"if in the numerator,"},{"Start":"04:30.600 ","End":"04:32.520","Text":"I have 1, that\u0027s good,"},{"Start":"04:32.520 ","End":"04:33.734","Text":"I have the formula."},{"Start":"04:33.734 ","End":"04:36.155","Text":"If I had 1 minus x squared,"},{"Start":"04:36.155 ","End":"04:38.675","Text":"that would also be good because then it would cancel,"},{"Start":"04:38.675 ","End":"04:40.910","Text":"so just messing around a bit,"},{"Start":"04:40.910 ","End":"04:45.410","Text":"I noticed that if I add the numerator to the denominator, I get 1."},{"Start":"04:45.410 ","End":"04:50.375","Text":"In other words, x squared plus 1 minus x squared,"},{"Start":"04:50.375 ","End":"04:53.165","Text":"I noticed it was equal to 1."},{"Start":"04:53.165 ","End":"04:55.745","Text":"If I have x squared,"},{"Start":"04:55.745 ","End":"05:00.920","Text":"that\u0027s just going to equal 1 minus,1 minus x squared"},{"Start":"05:00.920 ","End":"05:03.275","Text":"or you could have just gotten to this directly."},{"Start":"05:03.275 ","End":"05:07.730","Text":"In any event, what I\u0027m going to do is rewrite this into a form"},{"Start":"05:07.730 ","End":"05:10.100","Text":"where I have a difference rather than a quotient."},{"Start":"05:10.100 ","End":"05:11.420","Text":"The quotients is no good,"},{"Start":"05:11.420 ","End":"05:13.880","Text":"there\u0027s no quotient rule and integration."},{"Start":"05:13.880 ","End":"05:17.255","Text":"What I get is the integral,"},{"Start":"05:17.255 ","End":"05:19.810","Text":"now the denominator is the same,"},{"Start":"05:19.810 ","End":"05:21.570","Text":"integral sign is the same,"},{"Start":"05:21.570 ","End":"05:23.265","Text":"that dx is the same,"},{"Start":"05:23.265 ","End":"05:25.560","Text":"but what I\u0027m going to get here is a difference,"},{"Start":"05:25.560 ","End":"05:33.200","Text":"1 minus 1 minus x squared and I\u0027m going to highlight this minus"},{"Start":"05:33.200 ","End":"05:38.210","Text":"to show that we\u0027re going to break this up into 2 separate integrals."},{"Start":"05:38.210 ","End":"05:41.720","Text":"What we get is, first of all,"},{"Start":"05:41.720 ","End":"05:47.570","Text":"the integral of 1 over this minus the integral of this over this."},{"Start":"05:47.570 ","End":"05:56.795","Text":"The first one is 1 over 1 minus x squared dx and then minus the integral."},{"Start":"05:56.795 ","End":"06:00.530","Text":"Now, what\u0027s 1 minus x squared over 1 minus x squared?"},{"Start":"06:00.530 ","End":"06:04.940","Text":"That\u0027s exactly 1 dx and now I\u0027m in really good shape"},{"Start":"06:04.940 ","End":"06:07.580","Text":"because for the first part I have the formula"},{"Start":"06:07.580 ","End":"06:10.069","Text":"and the second part an immediate integral"},{"Start":"06:10.069 ","End":"06:15.410","Text":"so this comes out to be 1/2 just copying from over here,"},{"Start":"06:15.410 ","End":"06:17.225","Text":"just blind copying."},{"Start":"06:17.225 ","End":"06:23.790","Text":"Natural log of 1 plus x over 1 minus x"},{"Start":"06:23.790 ","End":"06:30.560","Text":"and of absolute value minus the integral of a constant is a constant times x,"},{"Start":"06:30.560 ","End":"06:32.990","Text":"so I have minus x and finally,"},{"Start":"06:32.990 ","End":"06:36.990","Text":"plus the constant, and it\u0027s the end of part c,"},{"Start":"06:36.990 ","End":"06:38.350","Text":"so that\u0027s the end."},{"Start":"06:38.350 ","End":"06:41.030","Text":"But I\u0027d like to mention that in future,"},{"Start":"06:41.030 ","End":"06:44.120","Text":"we won\u0027t have to pull formulas like this out of the hat."},{"Start":"06:44.120 ","End":"06:47.690","Text":"We\u0027d be able to solve these on our own using a technique"},{"Start":"06:47.690 ","End":"06:51.025","Text":"which is called decomposition into partial fractions,"},{"Start":"06:51.025 ","End":"06:53.420","Text":"but that\u0027s not for you to worry about now,"},{"Start":"06:53.420 ","End":"06:55.310","Text":"I\u0027m just telling you it\u0027s for the future."},{"Start":"06:55.310 ","End":"06:57.690","Text":"That\u0027s it."}],"ID":1506},{"Watched":false,"Name":"Exercise 16","Duration":"3m 29s","ChapterTopicVideoID":1517,"CourseChapterTopicPlaylistID":82571,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.com/Images/Videos_Thumbnails/1517.jpeg","UploadDate":"2017-01-25T23:54:05.7070000","DurationForVideoObject":"PT3M29S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:04.050","Text":"In this exercise, we have to compute the following 3 integrals: a,"},{"Start":"00:04.050 ","End":"00:07.905","Text":"b, and c. Let\u0027s start with a, which is a cosine."},{"Start":"00:07.905 ","End":"00:11.190","Text":"Normally the integral of cosine x is sine x,"},{"Start":"00:11.190 ","End":"00:16.275","Text":"but there\u0027s a more general rule with instead of x, ax plus b."},{"Start":"00:16.275 ","End":"00:20.580","Text":"In this case, we have to not just take the sign of what\u0027s in the brackets,"},{"Start":"00:20.580 ","End":"00:24.225","Text":"but divide by a or multiply by 1 over a,"},{"Start":"00:24.225 ","End":"00:27.735","Text":"and I\u0027ve indicated that a here as the 4."},{"Start":"00:27.735 ","End":"00:31.815","Text":"All I have to do here really is just use the formula."},{"Start":"00:31.815 ","End":"00:37.330","Text":"The integral of cosine is 1 over a, which is 1/4,"},{"Start":"00:37.330 ","End":"00:41.915","Text":"and sine of whatever was in the brackets for x,"},{"Start":"00:41.915 ","End":"00:45.625","Text":"and, finally, to add c. That\u0027s all there is to it."},{"Start":"00:45.625 ","End":"00:54.020","Text":"In Part b, I have the integral of sine of x over 2,"},{"Start":"00:54.020 ","End":"00:59.060","Text":"which I\u0027ll write as a 1/2x dx."},{"Start":"00:59.060 ","End":"01:02.810","Text":"Now, this is obviously the wrong set of formulae,"},{"Start":"01:02.810 ","End":"01:06.170","Text":"I\u0027ll need the ones with sine, so hang on."},{"Start":"01:06.170 ","End":"01:11.225","Text":"The formula I need here that the integral of sine is minus cosine,"},{"Start":"01:11.225 ","End":"01:12.900","Text":"but instead of x,"},{"Start":"01:12.900 ","End":"01:14.680","Text":"I have ax plus b,"},{"Start":"01:14.680 ","End":"01:18.470","Text":"then instead of the cosine of x plus b,"},{"Start":"01:18.470 ","End":"01:25.130","Text":"I have to also multiply by 1 over a or divide by a. I\u0027ll just highlight that."},{"Start":"01:25.130 ","End":"01:26.840","Text":"In other words, instead of x,"},{"Start":"01:26.840 ","End":"01:28.850","Text":"I have a times x,"},{"Start":"01:28.850 ","End":"01:30.605","Text":"and the a is from here,"},{"Start":"01:30.605 ","End":"01:34.550","Text":"then I also have to divide by a here,"},{"Start":"01:34.550 ","End":"01:39.935","Text":"and in our case, a is 1.5."},{"Start":"01:39.935 ","End":"01:43.370","Text":"In fact, we can also immediately write the solution,"},{"Start":"01:43.370 ","End":"01:45.680","Text":"because we do have like here,"},{"Start":"01:45.680 ","End":"01:46.835","Text":"where a is a 1/2,"},{"Start":"01:46.835 ","End":"01:48.290","Text":"b happens to be 0,"},{"Start":"01:48.290 ","End":"01:51.550","Text":"doesn\u0027t matter, so what I need is minus."},{"Start":"01:51.550 ","End":"01:55.620","Text":"Now, 1 over 1/2 is 2,"},{"Start":"01:55.620 ","End":"02:00.290","Text":"so it\u0027s minus 2 times sine of whatever was in the brackets,"},{"Start":"02:00.290 ","End":"02:02.260","Text":"which was a 1/2x,"},{"Start":"02:02.260 ","End":"02:05.150","Text":"but I\u0027ll write it back in the original form,"},{"Start":"02:05.150 ","End":"02:10.720","Text":"x over 2, and then plus c. That\u0027s it."},{"Start":"02:10.720 ","End":"02:16.070","Text":"What we have here is we can break this integral up into 2 integrals,"},{"Start":"02:16.070 ","End":"02:20.765","Text":"and we have because of this plus here and because we have a constant times,"},{"Start":"02:20.765 ","End":"02:26.455","Text":"we can write twice the integral of sine of"},{"Start":"02:26.455 ","End":"02:34.330","Text":"4x dx plus the integral of cosine x dx."},{"Start":"02:34.330 ","End":"02:37.940","Text":"Now, we have all the formulas we need."},{"Start":"02:37.940 ","End":"02:42.680","Text":"The sine of 4x is very similar to these exercises."},{"Start":"02:42.680 ","End":"02:47.495","Text":"We just have that our a is equal to this 4 here,"},{"Start":"02:47.495 ","End":"02:51.380","Text":"and we have to remember to multiply by 1 over a,"},{"Start":"02:51.380 ","End":"02:53.060","Text":"so we have twice,"},{"Start":"02:53.060 ","End":"02:55.910","Text":"times 1 over 4,"},{"Start":"02:55.910 ","End":"02:58.925","Text":"which is 1.5 times,"},{"Start":"02:58.925 ","End":"03:00.815","Text":"and because it\u0027s sine,"},{"Start":"03:00.815 ","End":"03:04.025","Text":"the integral is minus cosine,"},{"Start":"03:04.025 ","End":"03:06.925","Text":"so it\u0027s minus there,"},{"Start":"03:06.925 ","End":"03:12.525","Text":"times cosine of 4x plus."},{"Start":"03:12.525 ","End":"03:17.160","Text":"Now, the integral of cosine is just sine without the minus,"},{"Start":"03:17.160 ","End":"03:18.795","Text":"and it\u0027s just as is,"},{"Start":"03:18.795 ","End":"03:22.770","Text":"plus sine of x, and, finally,"},{"Start":"03:22.770 ","End":"03:25.630","Text":"plus c. That\u0027s it for Part c,"},{"Start":"03:25.630 ","End":"03:29.400","Text":"which means we\u0027ve finished this set of 3."}],"ID":1507}],"Thumbnail":null,"ID":82571}]

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1.1

1