1
00:00:00,25 --> 00:00:06,845
[MUSIC] We could figure out our anti 
derivative for log x, by just guessing 

2
00:00:06,845 --> 00:00:13,977
and then checking our answer. 
But we can also get that anti derivative 

3
00:00:13,977 --> 00:00:18,923
by using integration by parts. 
So I want to know an anti derivative of , 

4
00:00:18,923 --> 00:00:22,618
log x dx. 
And just to remind you, this is natural 

5
00:00:22,618 --> 00:00:25,487
log. 
At first, it looks like there's nothing 

6
00:00:25,487 --> 00:00:29,120
helpful here. 
What would my u and my dv be? 

7
00:00:29,120 --> 00:00:34,632
The trick is to sit u log x that's the 
part of the anagram I'm going to 

8
00:00:34,632 --> 00:00:41,711
differentiate as dv. 
To be dx and it gets quite surprising, 

9
00:00:41,711 --> 00:00:47,320
and that helps. 
Let's see, so if u is log x then du is 1 

10
00:00:47,320 --> 00:00:53,826
over x dx. 
And if dv is dx then an anti-derivative, 

11
00:00:53,826 --> 00:00:57,580
just be x. 
Now, what does parts tell us? 

12
00:00:57,580 --> 00:01:06,523
So by parts the integral of u dv, so log 
x dx, that's what I'm interested in. 

13
00:01:06,523 --> 00:01:14,125
Is u v so x log x minus the integral of 
vdu, which is x times du. 

14
00:01:14,125 --> 00:01:19,900
And I've got x log x minus x times 1 over 
x, that's just 1. 

15
00:01:19,900 --> 00:01:24,825
So I'm just anti-differentiating dx. 
Then I've got x log x, what's an 

16
00:01:24,825 --> 00:01:30,420
anti-derviative of 1, just x, and I'll 
add a constant. 

17
00:01:30,420 --> 00:01:34,934
So I'm claiming that the antiderivative 
of log x is x log x minus x plus some 

18
00:01:34,934 --> 00:01:39,824
constant. 
I think it's really surprising that, at 

19
00:01:39,824 --> 00:01:44,380
least in this case, dv equals dx is a 
great choice. 

20
00:01:44,380 --> 00:01:47,308
Well here's a similar integration problem 
that you can try the same kind of 

21
00:01:47,308 --> 00:01:50,500
technique on. 
For example, here's a challenge. 

22
00:01:50,500 --> 00:02:02,373
Can you find an anti derivative of log of 
x squared?