1 00:00:00,001 --> 00:00:08,881 [music] You've already seen some examples where we're antidifferentiating the powers 2 00:00:08,881 --> 00:00:11,619 of x. So here's an example. 3 00:00:11,620 --> 00:00:14,733 What would I have to differentiate, right? What goes in this cloud? 4 00:00:14,734 --> 00:00:19,822 So if I differentiate it I get 3 x squared or stated differently, what's the 5 00:00:19,822 --> 00:00:24,132 antiderivative of 3 x squared. I've just got to think of some function 6 00:00:24,132 --> 00:00:26,710 because if I differentiate it I get 3 x squared. 7 00:00:26,710 --> 00:00:31,714 Well I can think of one. X cubed is an example of a function that 8 00:00:31,714 --> 00:00:35,661 if I differentiate x cubed I get back 3 x squared. 9 00:00:35,661 --> 00:00:39,353 What about the general case? So when I think about what would I 10 00:00:39,353 --> 00:00:42,850 differentiate to get x to the n. Here I'm going to suppose that x is some 11 00:00:42,850 --> 00:00:46,763 number but it's not negative 1. Right, so in other words I want to know 12 00:00:46,763 --> 00:00:50,746 what the anti derivative of x to the n is. Right, so I'm thinking what would I 13 00:00:50,746 --> 00:00:55,345 differentiate to get x to the n? And, I'm going to claim that it's x to the 14 00:00:55,345 --> 00:00:59,289 n plus 1, divided by n plus 1. Now how do I know that this is the 15 00:00:59,289 --> 00:01:02,675 anti-derivative of x to the n? Well all I've got to do is differentiate. 16 00:01:02,675 --> 00:01:08,263 Alright, I'll differentiate x to the n plus 1 over n plus 1. 17 00:01:08,263 --> 00:01:11,619 And I'll do that with the power rule and the constant multiple rule. 18 00:01:11,620 --> 00:01:15,838 This constant multiple just comes on out of this differentiation. 19 00:01:15,838 --> 00:01:20,130 So I've got 1 over n plus 1 times the derivative of x to the n plus 1. 20 00:01:20,130 --> 00:01:25,122 Now, what does the power rule tell me? Power rule tells me how to differentiate 21 00:01:25,122 --> 00:01:26,135 this. Right? 22 00:01:26,135 --> 00:01:32,185 That's n plus 1, times x to the n plus 1 minus 1 which is just x to the n. 23 00:01:32,186 --> 00:01:37,971 But now this n plus 1 and n plus 1 cancels what I'm left with is just x to the n and 24 00:01:37,971 --> 00:01:43,667 that's exactly what I'm claiming right that the n type derivative of x to the n 25 00:01:43,667 --> 00:01:48,745 is x to the n plus 1 over n plus 1. I should be careful to point out something 26 00:01:48,745 --> 00:01:52,053 here. What if I wanted to anti differentiate a 27 00:01:52,053 --> 00:01:55,488 polynomial. For example let's suppose I want to find a 28 00:01:55,488 --> 00:01:59,113 function big f so that the derivative of big f is little f. 29 00:01:59,113 --> 00:02:06,394 And maybe little f is a polynomial like maybe little f is 15 x squared I don't 30 00:02:06,394 --> 00:02:10,625 know minus 4 x plus 3. All right so I'm really asking for an 31 00:02:10,625 --> 00:02:15,104 antiderivative of little f. I'm asking for a function whose derivative 32 00:02:15,104 --> 00:02:17,847 is this polynomial. We can totally do this. 33 00:02:17,848 --> 00:02:22,541 So let's do this using the notation that we've been developing. 34 00:02:22,542 --> 00:02:31,840 So I'm going to write antiderivative of 15 x squared minus 4 x plus 3 dx. 35 00:02:31,840 --> 00:02:35,580 And this is the antiderivative of a sum of the difference, and that's the sum of the 36 00:02:35,580 --> 00:02:41,084 difference of the antiderivative. So I can write this as the antiderivative 37 00:02:41,084 --> 00:02:46,745 of 15 x squared dx minus the anti-derivative of 4 x, dx plus the 38 00:02:46,745 --> 00:02:51,550 antiderivative of 3 dx. Now I've got the antiderivative of a 39 00:02:51,550 --> 00:02:56,168 number times something. So I can pull these numbers outside of the 40 00:02:56,168 --> 00:03:00,440 antiderivatives. So I rewrite this as 15 times the 41 00:03:00,440 --> 00:03:08,056 antiderivative of x squared dx minus four times the antiderivative of x dx, plus, 42 00:03:08,056 --> 00:03:12,361 alright, the antiderivative of 3 dx. Okay. 43 00:03:12,361 --> 00:03:15,630 Then I think what's the antiderivative of, of x squared. 44 00:03:15,630 --> 00:03:18,032 What's just an antiderivative of x squared? 45 00:03:18,032 --> 00:03:21,832 Well, I know one. It's x to the 3rd over 3, right? 46 00:03:21,832 --> 00:03:27,562 That's the power rule running in reverse. Minus 4 times, this is an antiderivative 47 00:03:27,562 --> 00:03:30,791 of x, something that I differentiate to get x. 48 00:03:30,792 --> 00:03:35,522 Well, that's x squared over 2, right? If I differentiate x squared over 2, I get 49 00:03:35,522 --> 00:03:37,415 back x. And what's an antiderivative of 3? 50 00:03:37,415 --> 00:03:41,955 What's something I differentiate to get 3? Well, 3x is such a thing, and to write 51 00:03:41,955 --> 00:03:46,344 down the most general antiderivative, I'm going to add plus C here. 52 00:03:46,345 --> 00:03:50,290 So this is the antiderivative of this polynomial. 53 00:03:50,290 --> 00:03:54,224 And I could write this a little bit more nicely now since some of these things 54 00:03:54,225 --> 00:03:59,220 cancel. 15 3rds is 5 x to the 3rd, minus 2 x 55 00:03:59,220 --> 00:04:05,391 squared, plus 3 x, plus c. Here we're using a sum rule, a power rule 56 00:04:05,391 --> 00:04:09,190 and also a constant multiple rule that we haven't seen yet. 57 00:04:09,190 --> 00:04:12,906 So it's worth writing down that constant multiple rule explicitly. 58 00:04:12,907 --> 00:04:16,666 Just remember what the constant multiple rule says for just differentiation. 59 00:04:16,666 --> 00:04:21,490 Yeah. If I differentiate say some number a times 60 00:04:21,490 --> 00:04:29,594 some function big f of x, well that's just a times the derivative of big f of x. 61 00:04:29,594 --> 00:04:34,432 And I can write down the same kind of rule, but for antiderivatives. 62 00:04:34,432 --> 00:04:40,105 For antiderivatives, if I'm antidifferentiating, say, a times little f 63 00:04:40,105 --> 00:04:45,111 of x. Well, this is the same as a times the 64 00:04:45,111 --> 00:04:50,960 antiderivative of f of x. Its not too difficult to justify this. 65 00:04:50,960 --> 00:04:57,745 So yeah lets try to justify this constant multiple rule for antiderivatives. 66 00:04:57,745 --> 00:05:02,920 Lets suppose that f differentiates to give me f of x. 67 00:05:02,920 --> 00:05:07,013 So I'm really is supposing that big f is an antiderivative for little f. 68 00:05:07,014 --> 00:05:14,254 Then this side here, I could rewrite a times the antiderivative of f as a times 69 00:05:14,254 --> 00:05:18,743 big f, it's an antiderivative of f, right, plus C. 70 00:05:18,744 --> 00:05:24,394 So that if I'm claiming that this, the antiderivative of a times f of x is this. 71 00:05:24,395 --> 00:05:29,760 It's enough to just check that the derivative of this is really this. 72 00:05:29,760 --> 00:05:32,997 And that's true, right? What's the derivative of a times F of x 73 00:05:32,997 --> 00:05:37,958 was exactly what I did up here, you know? These rules for antidifferentiation are 74 00:05:37,958 --> 00:05:40,927 exactly coming from the rules of differentiation. 75 00:05:40,928 --> 00:05:45,908 So the derivative of this constant multiple times F of x is that constant 76 00:05:45,908 --> 00:05:50,888 multiple times the derivative of F of x, right, and I'm claiming that the 77 00:05:50,888 --> 00:05:56,197 derivative of F of x is little f. So this is a times little f of x. 78 00:05:56,198 --> 00:06:00,720 Which is exactly saying that the antiderivative here, is this. 79 00:06:00,720 --> 00:06:09,803 Which is what I'm trying to justify.