1 00:00:00,25 --> 00:00:05,47 [MUSIC]. 2 00:00:05,47 --> 00:00:09,272 Sometimes the best substitution to make isn't even visible until after we've 3 00:00:09,272 --> 00:00:12,861 messed around with the integrand some how. 4 00:00:12,861 --> 00:00:20,172 For example, what's the antiderivative of 1 over 1 plus cosine x? 5 00:00:20,172 --> 00:00:24,721 Dx, what can you do? your first move might be to try the 6 00:00:24,721 --> 00:00:29,867 substitution u equals cosine x, which case du is negative sine x dx, but 7 00:00:29,867 --> 00:00:35,492 there's no visible sine x in the integrand. 8 00:00:35,492 --> 00:00:39,512 Well, since we're talking about invisible substitutions, we should try to mess 9 00:00:39,512 --> 00:00:44,143 around with the integrand. Instead of doing that, let's try a 10 00:00:44,143 --> 00:00:49,916 different trick. Let's try to multiply 1 over 1 plus 11 00:00:49,916 --> 00:00:59,240 cosine x by 1 minus cosine x divided by 1 minus cosine x. 12 00:00:59,240 --> 00:01:03,290 This doesn't change the integrand at all, because this is just 1. 13 00:01:03,290 --> 00:01:07,782 This trick makes a hitherto invisible substitution visible. 14 00:01:07,782 --> 00:01:12,100 I'm getting ahead of myself a little bit. Let's first apply a trig identity. 15 00:01:12,100 --> 00:01:19,524 Oh, this is 1 minus cosine x in the numerator divided by, that's 1 minus 16 00:01:19,524 --> 00:01:27,460 cosine squared x, and then the trig identity is that 1 minus cosine squared 17 00:01:27,460 --> 00:01:37,303 x, well that's sine squared x. So now I want to antidifferentiate 1 18 00:01:37,303 --> 00:01:42,530 minus cosine x over sine squared x dx. Well even that isn't so great. 19 00:01:42,530 --> 00:01:46,381 Let's split it up. Well then I get, that this is 20 00:01:46,381 --> 00:01:54,337 antiderivative 1 over sine squared x dx minus the antiderivative of cosine x over 21 00:01:54,337 --> 00:02:01,9 sine squared x dx. Now that first integral is one that I can 22 00:02:01,9 --> 00:02:05,250 do. Rewrite it as the antiderivative of 23 00:02:05,250 --> 00:02:11,8 cosecant squared x. You know, I just have to think, do I know 24 00:02:11,8 --> 00:02:16,357 any function whose derivative is cosecant squared. 25 00:02:16,357 --> 00:02:24,933 Yes, negative cotangent is an antiderivative of cosecant squared, of of 26 00:02:24,933 --> 00:02:29,0 x. What about that other intergral? 27 00:02:29,0 --> 00:02:33,331 Well I could read this as cotangent times cosecant and just recognize the 28 00:02:33,331 --> 00:02:38,443 antiderivative that way. But to demonstrate the technique I can 29 00:02:38,443 --> 00:02:42,987 also apply u substitution to that antidifferentiation problem so let's let 30 00:02:42,987 --> 00:02:47,176 u equal sine x, and in that case, du is cosine x dx, which is great, because 31 00:02:47,176 --> 00:02:55,815 that's the numerator there. So this this antidifferentiation problem 32 00:02:55,815 --> 00:03:01,251 becomes what? This is the antiderivative of du over u 33 00:03:01,251 --> 00:03:06,211 squared. And I just gotta think, how do I, 34 00:03:06,211 --> 00:03:13,190 antidifferentiate u to the negative second power? 35 00:03:13,190 --> 00:03:19,895 Well that, is by the power rule plus1 over u plus c. 36 00:03:19,895 --> 00:03:27,794 Alright, If I differentiate one over u, that gives me negative 1 over u squared. 37 00:03:27,794 --> 00:03:32,18 Okay, but I don't want my answer to be in terms of u, right, I want my answer to be 38 00:03:32,18 --> 00:03:37,20 in terms of x. So this is negative cotangent x plus, 39 00:03:37,20 --> 00:03:42,782 what's 1 over u, well that's 1 over sine x which, if I wanted to, I could write as 40 00:03:42,782 --> 00:03:49,60 cosecant x plus C. Let's put it all together. 41 00:03:49,60 --> 00:03:54,604 So what I'm claiming here is that the antiderivative of 1 over 1 plus cosine x 42 00:03:54,604 --> 00:04:01,640 dx is negative cotangent plus cosecant, plus some constant. 43 00:04:01,640 --> 00:04:08,140 If you have it already, I hope you're getting the idea that there's some really 44 00:04:08,140 --> 00:04:16,73 clever things that you can try to make these substitutions work.