1 00:00:00,25 --> 00:00:08,870 [MUSIC] Integrating requires patience. Sometimes you can start attacking the 2 00:00:08,870 --> 00:00:14,100 monster, but it might take a little bit of time before you start to see success. 3 00:00:14,100 --> 00:00:20,63 Prepare to persevere. Let's attack the indefinite integral of 4 00:00:20,63 --> 00:00:24,960 cosx times e to the x dx. There's a metaphor here. 5 00:00:24,960 --> 00:00:28,200 There's really different ways to win the battle. 6 00:00:28,200 --> 00:00:34,440 Suppose this is you, and this is the integral that you're trying to attack. 7 00:00:34,440 --> 00:00:37,330 One way, is you know, to try and do it by hand, right. 8 00:00:37,330 --> 00:00:40,466 To take your shield and to take your sword, and just to evaluate that integral 9 00:00:40,466 --> 00:00:45,110 directly by, say going back to the definition of Riemann sum. 10 00:00:45,110 --> 00:00:48,560 But we almost never attack integrals in this way. 11 00:00:48,560 --> 00:00:51,420 Instead of using, you know, swords and shields, we're really using, you know, 12 00:00:51,420 --> 00:00:52,650 magic. Right? 13 00:00:52,650 --> 00:00:55,683 We're putting on our wizard hat. And instead of trying to kill the 14 00:00:55,683 --> 00:00:59,252 intregral, right? We use something like u substitution to 15 00:00:59,252 --> 00:01:03,867 transform that integration problem into a friendly integration problem that we can 16 00:01:03,867 --> 00:01:08,784 actually deal with, right? We're never attacking integrals directly 17 00:01:08,784 --> 00:01:12,780 we're just transforming integrals by doing u substitutions or parts. 18 00:01:12,780 --> 00:01:17,70 So let's try to transform the integration that we're attacking here. 19 00:01:17,70 --> 00:01:20,880 Let's tranform e to the x cosx to something else. 20 00:01:20,880 --> 00:01:25,895 So I'll use what u, I'll make u be e to the x. 21 00:01:25,895 --> 00:01:28,480 because I don't really mind differentiating that. 22 00:01:28,480 --> 00:01:32,366 And I'll make dv cosx dx. Now in that case, the derivative of e to 23 00:01:32,366 --> 00:01:39,201 the x is just e to the x. And what's an anti-derivative of cosine? 24 00:01:39,201 --> 00:01:50,110 An anti-derivative of cosine is sine. Now what is integration by parts say. 25 00:01:50,110 --> 00:01:58,370 So then anti-derivative of cosx e to the x dx is U times V, so e to the x sinx 26 00:01:58,370 --> 00:02:08,530 minus the intergral of V du So, sinx, e to the x dx. 27 00:02:08,530 --> 00:02:11,980 Well that didn't really help matters. But the word of the day is persevere. 28 00:02:11,980 --> 00:02:17,556 So let's persevere, and transform that integral into yet another monster. 29 00:02:17,556 --> 00:02:21,30 So the monster in this case is this thing here. 30 00:02:21,30 --> 00:02:24,750 let's try the same trick on that new integral. 31 00:02:24,750 --> 00:02:31,491 So U will be e to the x again and dv here, will be just what's left over, sine 32 00:02:31,491 --> 00:02:37,487 x dx. And in that case, if U is e to the x, 33 00:02:37,487 --> 00:02:43,550 then du is e to the x dx. And, if dv is sin x dx, well, I've got 34 00:02:43,550 --> 00:02:49,648 other choices for more anti-derivative, but I'll have v be minus cosx. 35 00:02:49,648 --> 00:02:52,830 Now, I've put this into the integration by parts formula. 36 00:02:52,830 --> 00:03:03,125 So the antiderivative of sinx e to the x dx is uv, so minus e to the x cosx minus 37 00:03:03,125 --> 00:03:12,175 vdu. So my-, the integral of vdu, so v is 38 00:03:12,175 --> 00:03:21,635 cosx, and du is e to the x dx. Now what's happened thus far in the 39 00:03:21,635 --> 00:03:24,150 battle? I'm attacking this integral. 40 00:03:24,150 --> 00:03:28,740 I used integration by parts to transform the monster into something else. 41 00:03:28,740 --> 00:03:31,490 It didn't help, so I used that magical weapon again. 42 00:03:31,490 --> 00:03:34,626 I used integration by parts again to transform the integration problem into 43 00:03:34,626 --> 00:03:37,749 yet another integration problem and it didn't help. 44 00:03:38,870 --> 00:03:41,650 Or did it? What happens if I put these two 45 00:03:41,650 --> 00:03:46,434 integration facts together? Well, and then I've got here, the 46 00:03:46,434 --> 00:03:51,852 integral of cosx, e to the x dx, is equal to e to the x sin x minus the integral of 47 00:03:51,852 --> 00:04:01,810 sinx, e to the x, dx, which is down here. So minus negative e to x cos x plus the 48 00:04:01,810 --> 00:04:10,70 integral of cosine x e to the dx. Lets persevere, lets keep going. 49 00:04:10,70 --> 00:04:16,490 So this is saying the integral of cos x e to the x dx is e to the x sin x plus e to 50 00:04:16,490 --> 00:04:25,994 the x, cosine x, minus the integral of cosine x, e to the x, dx. 51 00:04:25,994 --> 00:04:32,809 Now I'm going to add the integral of cos x e to the x, dx to both sides. 52 00:04:32,809 --> 00:04:38,569 And what I'll then get is 2 times the integral of cos x e to the x, dx is equal 53 00:04:38,569 --> 00:04:44,233 to, well what's left on the righthand side, e to the x sin x plus e to the x, 54 00:04:44,233 --> 00:04:50,983 cos x. And then I'll divide both sides by 2, and 55 00:04:50,983 --> 00:04:55,527 if I divide this side by 2 and this side by 2 well then these 2's cancel, and I 56 00:04:55,527 --> 00:04:59,858 find out that an antiderivative of cosine x e to the x Is e to the x sin x plus z 57 00:04:59,858 --> 00:05:07,150 to the x cos x all over 2. What we did here is one of my favorite 58 00:05:07,150 --> 00:05:11,599 applications of integration by parts. We didn't use integration by parts to 59 00:05:11,599 --> 00:05:16,838 solve, to evaluate the integral directly. Instead, I applied parts twice and ended 60 00:05:16,838 --> 00:05:21,660 up with a clone of the original monster. So how did that go? 61 00:05:21,660 --> 00:05:25,116 Well, here you are, right, and you're a wizard, so here's your wizard hat, and 62 00:05:25,116 --> 00:05:28,419 here's the integral that we wanted to attack. 63 00:05:28,419 --> 00:05:30,322 But we didn't attack this integral directly. 64 00:05:30,322 --> 00:05:33,682 We didn't' even transform this integral by using U substitution. 65 00:05:33,682 --> 00:05:36,750 Instead, we used parts twice and it seemed like that was a terrible idea 66 00:05:36,750 --> 00:05:39,922 because when I applied parts twice, I ended up with a clone of the original 67 00:05:39,922 --> 00:05:43,552 monster. The same integral ended up appearing. 68 00:05:43,552 --> 00:05:46,992 But, that other integral came with a minus sign. 69 00:05:46,992 --> 00:05:50,712 And, consequently that monster ends up defeating itself and we're able to 70 00:05:50,712 --> 00:05:55,30 evaluate the integral without actually integrating at all. 71 00:05:55,30 --> 00:06:02,408 If anything suggests that there's creativity in the battles ahead of us, I 72 00:06:02,408 --> 00:06:12,333 think integration by parts is telling us, to be prepared to be amazed.