1 00:00:00,25 --> 00:00:05,8 [MUSIC]. 2 00:00:05,8 --> 00:00:08,573 Integrating is at its heart a creative process, right. 3 00:00:08,573 --> 00:00:12,437 Differentiating is just applying a bunch of rules carefully but just applying the 4 00:00:12,437 --> 00:00:15,998 rules. Integrating usually requires some sort of 5 00:00:15,998 --> 00:00:20,78 creative flash of insight. For u-substitution in particular, there's 6 00:00:20,78 --> 00:00:23,130 a ton of creativity in picking the best u. 7 00:00:23,130 --> 00:00:26,910 So if you're looking for how to pick u, right, what should you make u equal for 8 00:00:26,910 --> 00:00:31,623 your u substitution? Well, I'd look for things that you can 9 00:00:31,623 --> 00:00:34,937 grab as du, right? Try to find pieces of the integrand that 10 00:00:34,937 --> 00:00:38,894 look like the derivative of something. Well, let's try that. 11 00:00:38,894 --> 00:00:45,626 So, let's try the anti-differentiate, say, x over the square root of 4 minus 9x 12 00:00:45,626 --> 00:00:50,670 squared dx. So I can see that x squared in the 13 00:00:50,670 --> 00:00:53,610 denominator and I can see the x in the numerator. 14 00:00:53,610 --> 00:00:58,35 And I'm thinking, ahh, I put the x squared in the u so that the du will grab 15 00:00:58,35 --> 00:01:01,996 the x in the numerator. Let's go! 16 00:01:01,996 --> 00:01:09,180 So I proposed that u, should be 4 minus 9x squared. 17 00:01:09,180 --> 00:01:13,622 And then what's du? Well, calculate the differential of u by 18 00:01:13,622 --> 00:01:18,630 taking the derivative here and the derivative 4 is 0. 19 00:01:18,630 --> 00:01:24,30 But derivative negative 9x squared is negative 18x for the dx. 20 00:01:24,30 --> 00:01:26,30 And now you're thinking, oh, this is terrible. 21 00:01:26,30 --> 00:01:28,340 I don't see a negative 18 anywhere in this problem. 22 00:01:28,340 --> 00:01:32,778 But I can introduce one. I can make a negative 18 there as long as 23 00:01:32,778 --> 00:01:36,668 I cancel it with 1 over negative 18 there. 24 00:01:36,668 --> 00:01:40,850 So I've done nothing to the anti-differentiation problem. 25 00:01:40,850 --> 00:01:45,170 I haven't changed the problem at all, this is just multiplying by 1, they 26 00:01:45,170 --> 00:01:49,90 cancel. But I've now got a du in my integrand. 27 00:01:49,90 --> 00:01:54,38 So, let's make that substitution. This is now, well, the numerator is now 28 00:01:54,38 --> 00:01:59,854 just du, and the denominator is the square root of u. 29 00:01:59,854 --> 00:02:08,390 Now I want to make sure to include the 1 over negative 18 in front. 30 00:02:08,390 --> 00:02:11,110 And that integral I can do with the power rule. 31 00:02:11,110 --> 00:02:18,946 So this is, still 1 over negative 18, and what's an anti-derivative of this? 32 00:02:18,946 --> 00:02:23,682 Well this is the anti-derivative of u to the negative 1 half power, but that is 33 00:02:23,682 --> 00:02:28,344 exactly the sort of thing I can do with the power rule, right, so this is 1 over 34 00:02:28,344 --> 00:02:34,760 negative 18. Times by the power rule this is u to the 35 00:02:34,760 --> 00:02:39,480 1 half over 1 half plus C, which I could rewrite a little bit more nicely, 36 00:02:39,480 --> 00:02:44,120 dividing by 1 half is the same as multiplying by 2, so this is negative 1 37 00:02:44,120 --> 00:02:53,314 9th the square root of u plus C. Now I'll rewrite that in terms of x. 38 00:02:53,314 --> 00:02:59,199 So, in terms of x, this is, well remember, u is 4 minus 9x squared, so 39 00:02:59,199 --> 00:03:10,260 this is negative 1 9th the square root of 4 minus 9x squared, is what u is, plus C. 40 00:03:10,260 --> 00:03:17,610 So I found an anti-derivative of x over the square root of 4 minus 9x squared. 41 00:03:17,610 --> 00:03:21,900 I should warn you here, something that makes anti-differentiations so hard is 42 00:03:21,900 --> 00:03:26,255 that similar looking integration problems can have totally different looking 43 00:03:26,255 --> 00:03:29,855 answers. For example, we just saw how to 44 00:03:29,855 --> 00:03:35,328 anti-differentiate x over the square root of 4 minus 9x squared, we got this. 45 00:03:35,328 --> 00:03:39,423 How do we anti-differentiate this very similar looking function 1 over the 46 00:03:39,423 --> 00:03:43,518 square root of 4 minus 9x squared, the only difference is that I got rid of the 47 00:03:43,518 --> 00:03:48,360 x and the numerator. But that's a big difference. 48 00:03:48,360 --> 00:03:51,540 That x and the numerator was facilitating the substitution. 49 00:03:51,540 --> 00:03:56,650 I needed that x there in order to have something to fit into the du. 50 00:03:56,650 --> 00:04:00,114 Without that x there, what am I supposed to do? 51 00:04:00,114 --> 00:04:04,720 Well, I could try to factor out 2 from the denominator here. 52 00:04:04,720 --> 00:04:08,960 I mean, I'm not going to make a substitution yet, I'm just going to try 53 00:04:08,960 --> 00:04:14,400 to rewrite the integrand, so if I factor out a 2 this is the anti-derivative of 1 54 00:04:14,400 --> 00:04:21,906 over 2, times the square root of 1 minus 9 over 4x squared dx. 55 00:04:21,906 --> 00:04:26,556 And you might be wondering, you know, why is there a 2 on the outside and a 4 on 56 00:04:26,556 --> 00:04:31,409 the inside because the square root of 4 is 2. 57 00:04:31,409 --> 00:04:36,24 My plan here is to make the denominator look like the square root of, 1 minus 58 00:04:36,24 --> 00:04:42,483 something squared. Rewrite this, integral again as 1 over 2, 59 00:04:42,483 --> 00:04:49,373 the square root of 1 minus, and instead of 9 quarters x squared, I'll write it as 60 00:04:49,373 --> 00:04:56,100 3 halves x squared dx. And now I'll make a u-substitution. 61 00:04:56,100 --> 00:05:06,590 I'll say u equals to this 3 halves x. So, 3 halves x so that du is 3 halves dx. 62 00:05:06,590 --> 00:05:09,400 And now you're going to complain. Well, I don't see a 3 halves dx. 63 00:05:09,400 --> 00:05:14,298 Then, you do have a 1 half dx here, and I can manufacture a 3 here as long as I'm 64 00:05:14,298 --> 00:05:21,264 willing to put a 1 3rd on the outside that doesn't affect anything. 65 00:05:21,264 --> 00:05:25,36 But now I've got a 3 dx over 2, so I've got a du. 66 00:05:25,36 --> 00:05:28,316 1 over the square root of 1 minus u squared. 67 00:05:28,316 --> 00:05:33,115 Let's write that down. So this whole thing is 1 3rdthe 68 00:05:33,115 --> 00:05:40,738 anti-derivative of 1 over the square root of 1 minus, this is u squared, and 3 69 00:05:40,738 --> 00:05:47,360 halves dx is du. Why is this a good idea? 70 00:05:47,360 --> 00:05:51,638 Well this is a good idea because this is now 1 3rd, and I know something which 71 00:05:51,638 --> 00:05:56,837 differentiates to this. Arc sin u, alright, is the 72 00:05:56,837 --> 00:06:02,472 anti-derivative of this. To finish this, I'll replace u by 3 73 00:06:02,472 --> 00:06:08,422 halves x. So I get that this is 1 3rd arcsin of 3 74 00:06:08,422 --> 00:06:12,910 halves x plus C. Alright? 75 00:06:12,910 --> 00:06:16,894 This is just 'cuz you is 3 halves x. Look at how different this experience 76 00:06:16,894 --> 00:06:19,795 was. Compared to that first integral that we 77 00:06:19,795 --> 00:06:23,5 did. Just showed that the anti-derivative of 1 78 00:06:23,5 --> 00:06:27,76 over the square root of 4 minus 9x squared is 1 3rd the arcsin of 3 halves x 79 00:06:27,76 --> 00:06:31,531 plus C. And that looks totally different from 80 00:06:31,531 --> 00:06:36,986 anti-differentiating x over the square root of 4 minus 9x squared, right? 81 00:06:36,986 --> 00:06:40,316 I didn't need any inverse trig functions there. 82 00:06:40,316 --> 00:06:45,170 You know even though the integrands look relatively similar, right? 83 00:06:45,170 --> 00:06:50,120 I'm mean I'm just missing an x but it totally transforms the answer. 84 00:06:50,120 --> 00:06:54,720 Picking the best u for your u-substitutions really is an art form. 85 00:06:54,720 --> 00:06:58,690 This is why people have integration bees just like they have spelling bees, right? 86 00:06:58,690 --> 00:07:09,509 There's a real creativity to integrating.