1 00:00:00,25 --> 00:00:05,21 [MUSIC]. 2 00:00:05,21 --> 00:00:08,773 In order to integrate a complicated function, I really just want to 3 00:00:08,773 --> 00:00:15,671 antidifferentiate complicated functions. For example, can I antidifferentiate x 4 00:00:15,671 --> 00:00:20,869 times sin of x squared dx? And if you think back to when we were 5 00:00:20,869 --> 00:00:25,930 doing all our differentiation stuff, a big deal was the chain rule. 6 00:00:25,930 --> 00:00:30,120 We were always using the chain rule in order to differentiate things. 7 00:00:30,120 --> 00:00:33,970 So for this problem I might try to think in terms of the chain rule, right? 8 00:00:33,970 --> 00:00:39,680 I know some function who's derivative is sin, minus cosine differentiates to sin. 9 00:00:39,680 --> 00:00:43,896 But, and I want an x squared in there somehow, so I'll put in x squared there, 10 00:00:43,896 --> 00:00:48,400 and I'll see, is that an antiderivative of this? 11 00:00:48,400 --> 00:00:51,960 Well, let's try it. So if I differentiate this, what do I 12 00:00:51,960 --> 00:00:54,743 get? Well the derivative of the outside 13 00:00:54,743 --> 00:00:58,964 function is sin, evaluated the inside function times the derivative of the 14 00:00:58,964 --> 00:01:03,94 inside function, which in this case is 2x. 15 00:01:03,94 --> 00:01:08,0 And I see whoops, I'm off, right? I didn't quite get an antiderivative. 16 00:01:08,0 --> 00:01:11,654 I'm off by this factor of 2, so I can fix that, I'll just divide this by 2, which 17 00:01:11,654 --> 00:01:15,482 will have the effect of dividing that by 2, but then these 2s will cancel, and now 18 00:01:15,482 --> 00:01:21,870 I have found an antiderivative for x times sin of x squared. 19 00:01:21,870 --> 00:01:26,30 And yeah, that works, but it was totally adhoc, I mean how did I know to divide by 20 00:01:26,30 --> 00:01:32,36 2, I just guessed and fixed my guess. Mathematics really shouldn't be seen as 21 00:01:32,36 --> 00:01:36,245 just a series of tricks, a big part of mathematics is systematizing those 22 00:01:36,245 --> 00:01:42,710 tricks, finding the patterns that unify, trick into a tool. 23 00:01:42,710 --> 00:01:46,415 In this case, we really want to systematize applying the chain rule in 24 00:01:46,415 --> 00:01:49,908 reverse. So this process of applying the chain 25 00:01:49,908 --> 00:01:53,810 rule in reverse, goes by the name u substitution. 26 00:01:53,810 --> 00:01:57,779 It's also just called substitution, but I'm going to call it u substitution to 27 00:01:57,779 --> 00:02:02,323 emphasize the conventional name u. That I'm going to use for the inside 28 00:02:02,323 --> 00:02:05,767 function when we're running the chain rule backwards. 29 00:02:05,767 --> 00:02:09,660 This is, any how, all too abstract, let's just see this in action. 30 00:02:09,660 --> 00:02:16,840 So I'm trying to anti differentiate x times sin of x squared dx. 31 00:02:16,840 --> 00:02:20,620 And the trick here, is to give a name to the inside function in the chain rule, 32 00:02:20,620 --> 00:02:25,370 I'm going to call that u. And I want the inside function to be x 33 00:02:25,370 --> 00:02:30,525 squared, so I'll say that u is x squared. Well then, what's du? 34 00:02:30,525 --> 00:02:36,985 Right, what's the differential of u? Well, du over dx is the derivative, which 35 00:02:36,985 --> 00:02:41,292 is 2x, so du is 2x dx. I know you might feel kind of bad, 36 00:02:41,292 --> 00:02:45,245 because well I don't really see a 2x dx, I only see an x dx, but this sort of 37 00:02:45,245 --> 00:02:50,80 method going to, guides us to do the right thing. 38 00:02:50,80 --> 00:02:53,740 I'd like to have an 2x dx so I could put a 2 here, as long as I'm willing to put a 39 00:02:53,740 --> 00:02:58,80 1 half on the outside. It's like doing nothing. 40 00:02:58,80 --> 00:03:02,428 But now I've got a 2x dx in the integrand, and that'll become my du. 41 00:03:02,428 --> 00:03:07,388 So this antidifferentiation problem is the same as sin u du, and I've got to 42 00:03:07,388 --> 00:03:12,828 make sure to include that one half on the outside, but now I know an antiderivative 43 00:03:12,828 --> 00:03:19,370 for sin of u, right, it's negative cosine. 44 00:03:19,370 --> 00:03:25,866 So I've got one half and then an anti-derivative of sin of u is negative 45 00:03:25,866 --> 00:03:31,948 cosin plus c. But I don't want my answer to be in terms 46 00:03:31,948 --> 00:03:37,466 of u so I rewrite this as negative one half cosin of u, which is x squared plus 47 00:03:37,466 --> 00:03:42,560 c. There's something to notice here. 48 00:03:42,560 --> 00:03:46,281 I'm using differential with the chain rule so that dx is playing a crucial 49 00:03:46,281 --> 00:03:49,237 role. You might have thought that I was just 50 00:03:49,237 --> 00:03:52,828 writing dx at the end of my integration problem out of habit or tradition but 51 00:03:52,828 --> 00:03:58,216 that dx is legitimately there. That dx is managing the substitution for 52 00:03:58,216 --> 00:04:00,989 us. Let's see why this work? 53 00:04:00,989 --> 00:04:06,203 Every differentiation rule has a corresponding anti-differentiation rule, 54 00:04:06,203 --> 00:04:08,587 right? Let's say that I want to 55 00:04:08,587 --> 00:04:12,568 anti-differentiate f prime of g of x times g prime of x. 56 00:04:12,568 --> 00:04:16,686 Well, secretly I can recognize this is the derivative of f of g of x with 57 00:04:16,686 --> 00:04:20,606 respect to x. But let's suppose that I start writing it 58 00:04:20,606 --> 00:04:25,451 down using this substitution framework. So I'm making a substitution u. 59 00:04:25,451 --> 00:04:32,890 Is g of x, and in that case du is the derivative of g dx. 60 00:04:32,890 --> 00:04:39,652 So this antidifferentiation problem is the antidifferentiation problem f prime 61 00:04:39,652 --> 00:04:44,355 of u, and this now du. But I know an antiderivative of a 62 00:04:44,355 --> 00:04:48,940 derivative is just the original function. And in this case, u is g of x, so this is 63 00:04:48,940 --> 00:04:52,695 f of g of x. I don't know, 'cuz I mean I can really 64 00:04:52,695 --> 00:04:57,659 see this working, you know the derivative of this composition is this, I mean it's 65 00:04:57,659 --> 00:05:03,76 just the chain rule. We're going to see a ton more examples of 66 00:05:03,76 --> 00:05:07,52 this technique, and we're going to see that the hard part boils down to 67 00:05:07,52 --> 00:05:15,321 determining what to set u equal to. But if you're ever wondering why does use 68 00:05:15,321 --> 00:05:23,505 substitution work, remember, it's just a luren neock, I mean use substition is 69 00:05:23,505 --> 00:05:31,216 luren neock, a chain rule but in reverse.