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These are the user uploaded subtitles that are being translated: 1 00:00:00,320 --> 00:00:05,190 Today we're going to talk about how to use integration by parts three times in order to find the antiderivative 2 00:00:05,250 --> 00:00:10,980 of a function to complete this problem will understand our integration by parts formula and then use 3 00:00:10,980 --> 00:00:15,130 it three times in a row to simplify and evaluate the integral. 4 00:00:15,330 --> 00:00:22,190 In this particular problem we've been asked to evaluate the integral of x cube times either the x dx. 5 00:00:22,200 --> 00:00:28,500 The first thing that we should recognize is that we have essentially two functions inside of arm integral. 6 00:00:28,500 --> 00:00:36,450 Here we have x cubed and we have e to the X and they are multiplied together. 7 00:00:36,450 --> 00:00:42,400 So the first thing that we should think is to try to use integration by parts to evaluate this integral. 8 00:00:42,420 --> 00:00:47,040 Whenever you have two functions like this and they're multiplied together integration by parts should 9 00:00:47,040 --> 00:00:48,620 be your first thought. 10 00:00:48,720 --> 00:00:54,030 So I'd gone ahead and written the integration by parts formula over here on the right and what it tells 11 00:00:54,030 --> 00:01:00,000 us is that when we have a a when we have an integral when we are when we're asked to take the integral 12 00:01:00,420 --> 00:01:06,540 of two pieces that are multiplied together you and DVH when we want to take the integral of something 13 00:01:06,540 --> 00:01:14,030 in that form the formula will use to do it is you Times V minus the integral of V times d u. 14 00:01:14,310 --> 00:01:20,340 So essentially what we need to do is we need to identify which part of our integral is you and which 15 00:01:20,340 --> 00:01:21,730 part is DVH. 16 00:01:22,110 --> 00:01:29,430 And then once we've identified you MTV will take the derivative of you to find the you and the integral 17 00:01:29,490 --> 00:01:38,130 of DVH to find V and then we have u d u V and D and those four components we can use to plug into our 18 00:01:38,130 --> 00:01:43,290 formula and build out the right hand side over here. 19 00:01:43,320 --> 00:01:48,990 So let's go ahead first and identify you and DVH you. 20 00:01:49,110 --> 00:01:52,490 Is the part you want to focus in on first. 21 00:01:52,530 --> 00:01:59,680 So to find you you should be looking for something that becomes simpler when you take its derivative. 22 00:01:59,700 --> 00:02:05,580 So in this case we have X to the third and E to the x which are two candidates. 23 00:02:05,580 --> 00:02:11,700 If we take the derivative of each of the X we're going to get it the X the exact same thing and we really 24 00:02:11,700 --> 00:02:16,300 haven't made any progress which should immediately rule it out as a candidate for you. 25 00:02:16,380 --> 00:02:21,190 That means that you will be equal to x to the third. 26 00:02:21,210 --> 00:02:22,710 The other option. 27 00:02:22,770 --> 00:02:25,940 So we set you equal to x cubed. 28 00:02:25,980 --> 00:02:32,550 That means that the rest of what's inside the integral here has to be set equal to dvh because we have 29 00:02:33,000 --> 00:02:37,800 two components here we have you and we have DVH. 30 00:02:37,850 --> 00:02:44,900 So because we already set you equal to x cube that means that divi has to be equal to everything else 31 00:02:45,050 --> 00:02:46,450 including the X. 32 00:02:46,490 --> 00:02:53,020 So we'll set divi equal to each of the X Dyaks. 33 00:02:53,030 --> 00:02:57,950 Now as I mentioned before now that we've identified these two pieces we need to go ahead and take the 34 00:02:57,950 --> 00:03:00,950 derivative of you to get the U. 35 00:03:01,100 --> 00:03:06,020 So the derivative of x cubed is 3 x squared. 36 00:03:06,020 --> 00:03:11,050 And it's important then that we add a D X to this since we took the derivative. 37 00:03:11,360 --> 00:03:20,060 We're going to take the integral of DV to find V and the integral of the x is simply E to the X.. 38 00:03:20,060 --> 00:03:21,250 The same thing. 39 00:03:21,290 --> 00:03:26,630 So here with you you're going in this direction taking the derivative with divi you're going in this 40 00:03:26,630 --> 00:03:29,830 direction and taking the integral. 41 00:03:29,930 --> 00:03:35,780 Now that we have those four components we can go ahead and plug them into our integration by parts formula. 42 00:03:35,780 --> 00:03:39,130 So you'll notice that we have first you Times V. 43 00:03:39,340 --> 00:03:41,750 So going to be multiplying you and B together. 44 00:03:41,870 --> 00:03:51,130 So go ahead and say that are integral x cubed either the x dx our original integral is equal to this 45 00:03:51,130 --> 00:03:52,380 right hand side here. 46 00:03:52,420 --> 00:04:01,750 So you times being we know to be x cubed times each of the X minus the integral of V times you saw say 47 00:04:01,750 --> 00:04:14,260 the integral of V times d you the times c you will give us 3 x squared E to the x dx and now you can 48 00:04:14,260 --> 00:04:19,250 see we still have an integral that's too complicated to integrate on its own. 49 00:04:19,540 --> 00:04:24,020 But it is different than what we originally started with instead of having executed. 50 00:04:24,040 --> 00:04:25,980 We've got 3 x squared. 51 00:04:26,080 --> 00:04:30,490 So notice that the degree of that term went down by 1. 52 00:04:30,490 --> 00:04:36,600 Right we had to do a third degree power function here and now we have a second degree power function. 53 00:04:36,610 --> 00:04:42,460 So what we're going to have to do is use integration by parts again to try to attempt to reduce our 54 00:04:42,460 --> 00:04:44,290 integral to something simpler. 55 00:04:44,290 --> 00:04:54,250 So we'll again say this time that you is equal to 3 x squared we'll say that DVM is equal to everything 56 00:04:54,250 --> 00:04:56,930 that's left over which will be e to the X. 57 00:04:56,940 --> 00:05:01,830 The x will take the derivative of you to get d u. 58 00:05:02,050 --> 00:05:10,150 And that will give us 6 x times DX right we just used power rule to find the derivative of 3 x squared. 59 00:05:10,210 --> 00:05:18,640 So 6 x x and we'll take the integral of DVH the integral of each of the x is simply E to the X and now 60 00:05:18,640 --> 00:05:24,430 we'll plug these four components into our formula again for this integrals so what we're left with here 61 00:05:24,880 --> 00:05:32,110 is is x cubed either the X Kari's down so x cubed into the X minus. 62 00:05:32,110 --> 00:05:39,670 And now this integration by parts operation we're going to perform represents this integral right here. 63 00:05:39,670 --> 00:05:43,040 So what we'll do is we carry down our minus sign here. 64 00:05:43,120 --> 00:05:49,060 We're going to draw a big parentheses and plug these four components into the right hand side of our 65 00:05:49,060 --> 00:05:52,750 integration by parts formula and put that in parentheses here. 66 00:05:52,750 --> 00:06:04,720 So you at times V will be 3 x squared each to the X minus the integral V times d u which will give us 67 00:06:05,260 --> 00:06:09,310 6 x to the x x. 68 00:06:09,310 --> 00:06:11,590 Right we have the 6 x here. 69 00:06:11,750 --> 00:06:16,500 The either the X here and the X here are just reordered the terms like that. 70 00:06:16,780 --> 00:06:23,080 So that's our entire integration by parts formula is right hand side over here that replaced this integral 71 00:06:23,080 --> 00:06:24,680 right here. 72 00:06:24,700 --> 00:06:35,590 So when we simplify this we'll get x cubed into the X minus 3 x squared E to the X the negative sign 73 00:06:35,590 --> 00:06:39,900 here and here will cancel out and we'll get a positive. 74 00:06:39,910 --> 00:06:47,830 Now we can go ahead and pull this 6 here out in front of our integral because it's a constant coefficient 75 00:06:47,830 --> 00:06:53,950 so we can say plus six times the integral of x E to the x dx. 76 00:06:53,950 --> 00:06:58,730 It's not really crucial that we pull that 6 out in front but we can do it just a simple fire integral 77 00:06:58,750 --> 00:06:59,530 a little bit. 78 00:06:59,530 --> 00:07:05,920 So now notice that we still have a function that is too complicated for us to integrate as is will in 79 00:07:05,920 --> 00:07:07,590 fact need to use integration by parts. 80 00:07:07,600 --> 00:07:08,890 A third time. 81 00:07:08,890 --> 00:07:15,840 But the trend that we should take note of is that we started out with a third degree power function 82 00:07:15,840 --> 00:07:16,450 here. 83 00:07:16,750 --> 00:07:23,740 We use integration by parts and we reduce it to a second degree power function and now essentially we 84 00:07:23,740 --> 00:07:28,910 have X to the first power which is of course a first degree power function. 85 00:07:28,930 --> 00:07:30,910 So we've gone from three to two to one. 86 00:07:30,910 --> 00:07:37,600 So if the trend holds and we use integration by parts again this term here should drop out and we should 87 00:07:37,600 --> 00:07:41,190 be left with something that we can integrate. 88 00:07:41,500 --> 00:07:49,960 So if we use integration by parts again and we say that you is equal to x and that DVN is equal to each 89 00:07:49,960 --> 00:07:58,010 of the x dx and we take the derivative of you the derivative of X is just one. 90 00:07:58,150 --> 00:08:05,220 So we would get 1 times DX which of course would just be the X and then we integrate divi to find. 91 00:08:05,230 --> 00:08:06,640 So we get we equals. 92 00:08:06,870 --> 00:08:08,140 It is the X. 93 00:08:08,140 --> 00:08:12,970 Now plug those four components into our integration by parts formula to replace the integral that we 94 00:08:12,970 --> 00:08:14,220 had left over here. 95 00:08:14,470 --> 00:08:21,610 So what we get for our integral function will carry over this excuse me the x this whole thing here 96 00:08:21,610 --> 00:08:30,250 will bring up so x cubed you to x minus 3 x squared each x plus 6. 97 00:08:30,250 --> 00:08:35,570 And now here's where we draw our big parentheses and in place of this entire integral right here or 98 00:08:35,580 --> 00:08:41,710 place that with our the right hand side here of our integration by parts formula plugging in these components 99 00:08:41,710 --> 00:08:42,780 that we have. 100 00:08:42,790 --> 00:08:44,960 So remember you Times V. 101 00:08:45,010 --> 00:08:54,720 So you Times V will give us X E to the X minus the integral of V times d u. 102 00:08:54,730 --> 00:09:05,670 So v times d you here will just give us e to the x dx and notice now that our x term dropped away and 103 00:09:05,670 --> 00:09:09,360 we're just left with either the ex insider integral so it looks like we're finally going to be able 104 00:09:09,360 --> 00:09:13,710 to integrate after using integration by parts for the third time. 105 00:09:13,710 --> 00:09:24,270 So we'll go out and simplify and we'll get x cubed to the X minus 3 x Square-D to the X plus 6 x 8 to 106 00:09:24,270 --> 00:09:25,440 the X.. 107 00:09:25,470 --> 00:09:27,900 Notice now we have to distribute 6. 108 00:09:27,900 --> 00:09:34,160 So we'll get minus six times the integral of either the X dx. 109 00:09:34,230 --> 00:09:42,720 We know that the integral of E to the x is simply E to the X. So I'll go ahead and take the integral 110 00:09:43,710 --> 00:09:49,520 and we'll get 6 x either the X minus six times each to the X. 111 00:09:49,710 --> 00:09:56,310 At this point because we finished our final integral we want to make sure to add the constant of integration 112 00:09:56,550 --> 00:10:00,870 plus C to account for that that constant that could have dropped off. 113 00:10:00,870 --> 00:10:07,040 So we have the constant integration there and now we could leave this as our final answer. 114 00:10:07,140 --> 00:10:12,330 But why don't we go ahead and factor out in either the X because we have an either the X and each one 115 00:10:12,330 --> 00:10:18,180 of our terms here and we'll simplify if we factor out the either the X so our final answer for the integral 116 00:10:18,570 --> 00:10:24,530 will be either the X times x cubed minus 3 x squared. 117 00:10:25,840 --> 00:10:30,460 Plus 6 X minus 6 plus C. 13338