1 00:00:00,25 --> 00:00:04,956 [MUSIC]. 2 00:00:04,956 --> 00:00:09,180 Sometimes you might want to do u substitution more than once. 3 00:00:09,180 --> 00:00:13,124 Imagine that you've got something just really terrible-looking, like trying to 4 00:00:13,124 --> 00:00:17,311 find an anti-derivative of this, negative 2 cosine x sin x. 5 00:00:17,311 --> 00:00:21,27 Cosine, cosine squared x plus 1 dx, right? 6 00:00:21,27 --> 00:00:25,60 Predefine an anti-derivative of this. We can. 7 00:00:25,60 --> 00:00:29,960 You might be thinking ugh, if I could just get rid of that cosine squared term. 8 00:00:29,960 --> 00:00:34,50 Well to do that, let's say u, equaled a cosine x. 9 00:00:34,50 --> 00:00:37,914 Let's write that down, so the substitution that I'm proposing, is u 10 00:00:37,914 --> 00:00:41,792 equals cosine x. And that'll be good cause I've got a 11 00:00:41,792 --> 00:00:45,698 cosine x and a cosine squared x here and I can also see du in this intergrand, 12 00:00:45,698 --> 00:00:48,460 right? What's du? 13 00:00:48,460 --> 00:00:52,472 Well it's the differential of u. What's the derivative of cosine? 14 00:00:52,472 --> 00:00:56,688 It's minus sine, so the differential, then includes this dx, right? 15 00:00:56,688 --> 00:01:00,224 Du is negative sine x dx. Well let's make the substitution. 16 00:01:00,224 --> 00:01:05,86 Let's see how it works out. So this, anti-derivative problem is now, 17 00:01:05,86 --> 00:01:10,101 with a minus sin X DX, or become the DU but I'm left with a two cos X as U, the 18 00:01:10,101 --> 00:01:14,691 minus two and the sin X and the DX is going to be in the DU, but I got a cos 19 00:01:14,691 --> 00:01:20,131 sign, cos sign squared X is U squared plus one and everything else that's left 20 00:01:20,131 --> 00:01:29,140 over Is in this du. Now I'll make another substitution. 21 00:01:29,140 --> 00:01:33,540 I'll make v equal u squared plus 1. Well let's write that down. 22 00:01:33,540 --> 00:01:39,940 So the substitution that I'm proposing is v equals u squared plus 1. 23 00:01:39,940 --> 00:01:43,594 And that's a good choice, because I've got a U squared plus 1 there, and I can 24 00:01:43,594 --> 00:01:47,689 see that the derivative of this appears here so this will grab quite a bit of the 25 00:01:47,689 --> 00:01:53,286 integrant. Okay, let's site then the differential, 26 00:01:53,286 --> 00:01:58,26 DV in that case is the derivative to you DU so DV is 2U, DU and now I can write 27 00:01:58,26 --> 00:02:03,319 down what this anti-differentiation problem becomes 2u du ends up being the 28 00:02:03,319 --> 00:02:10,540 the dv. And I've got a cosine of just v now. 29 00:02:10,540 --> 00:02:15,173 Now I can anti-differentiate that no problem. 30 00:02:15,173 --> 00:02:19,940 Right. The anti-derivative of cosine v dv. 31 00:02:19,940 --> 00:02:24,386 Is just, sine v. I'll write plus c, right? 32 00:02:24,386 --> 00:02:27,576 because the derivative of sine v is cosine v with respect to v, but probably 33 00:02:27,576 --> 00:02:30,821 the grader is not going to be too impressed with me if I express my answer 34 00:02:30,821 --> 00:02:35,620 in terms of a substitution that I just made up, right? 35 00:02:35,620 --> 00:02:38,596 I shouldn't write down the answer in terms of v, I should be writing down the 36 00:02:38,596 --> 00:02:43,338 answer in terms of x. Remember from before that v is u squared 37 00:02:43,338 --> 00:02:49,58 plus 1, so I can subsitute that in here, and instead of writing down sin of v plus 38 00:02:49,58 --> 00:02:58,378 C, I'll write down sin of u squared plus 1, and then I'll include that plus C. 39 00:02:58,378 --> 00:03:03,380 Da, my answer is in terms of u, but the original question was in terms of x. 40 00:03:03,380 --> 00:03:06,910 Let's remember what u was. u was cosine x so I can make that final 41 00:03:06,910 --> 00:03:11,198 substitution back in here. So instead of writing sine of u squared 42 00:03:11,198 --> 00:03:15,876 plus 1. I'll write down sine of cosine squared x, 43 00:03:15,876 --> 00:03:21,840 because u is cosine x. Plus 1, plus C. 44 00:03:21,840 --> 00:03:25,865 So what am I claiming? The original question was asking for an 45 00:03:25,865 --> 00:03:30,350 anti derivative of this crazy expression, and what I'm claiming, then, is that the 46 00:03:30,350 --> 00:03:36,84 anti derivative of this is sign of cosign squared X plus 1 plus C. 47 00:03:36,84 --> 00:03:39,288 So we did it. And I bet that you can imagine some truly 48 00:03:39,288 --> 00:03:43,644 terrible anti differentiation problems that involved substitutions within 49 00:03:43,644 --> 00:03:48,980 substitutions within substitutions. Part of the problem with this calculus 50 00:03:48,980 --> 00:03:52,665 course, I suppose with calculus courses generally, is that the courses are sort 51 00:03:52,665 --> 00:03:58,350 of based on this idea of the instructor just providing problems to you. 52 00:03:58,350 --> 00:04:01,950 Here's the challenge. Invent some difficult antiderivative 53 00:04:01,950 --> 00:04:05,684 problems for yourself. By cooking up your own 54 00:04:05,684 --> 00:04:12,132 antidifferentiation problems, you'll gain some insight into what makes 55 00:04:12,132 --> 00:04:19,412 antidifferentiation problems easy, or what makes antidifferentiation problems 56 00:04:19,412 --> 00:04:22,256 hard.