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[MUSIC]. 

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The basic idea of integration by parts is 
that it lets you differentiate part of 

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the integrand, but only if you're willing 
to pay a price. 

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And that price is anti-differentiating 
the other part of the integrand. 

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Well, let's try it. 
For example, let's find an 

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anti-derivative of x times e to the x. 
If we're going to attack this integrand 

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by parts, then I've got to pick a u and a 
dv. 

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I'd be willing to differentiate x. 
That'll make x just go away. 

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And the price I'd have to pay is to 
anti-differentiate what remains. 

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But anti-differentiating e to the x is 
not much of a price to pay because e to 

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the x is its own anti-derivative. 
So let's set that down. 

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Let's set u to the x, because I'd like to 
differentiate that part. 

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And I'm willing to anti-differentiate 
what remains. 

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Now, let's figure out du and v. 
Now if u is x, then du is dx. 

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And then, I've got to pick an 
anti-derivative for dv. 

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Now, in principle, there's a ton of 
anti-derivatives I could pick, right? 

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e to the x plus 17 differentiates the e 
to the x. 

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But I'll just pick the nice one. 
I'll pick e to the x. 

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Now I've got u, dv, v, and du. 
We can put it all together. 

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Parts tells me that an anti-derivative u 
dv is uv minus anti-derivative v du. 

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So, in this case, the anti-derivative of 
u dv is u times v, x times e to the x 

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minus anti-derivative of v du, which is 
just dx. 

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But, I know how to integrate e to the x. 
Anti-derivative of e to the x is just 

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itself. 
So, the anti-derivative of x, e to the x 

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is x, e to the x minus e to the x plus 
some constant. 

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We did it, and we can check our answer. 
So we have to differentiate this and make 

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sure I get x, e to the x. 
Let's try it. 

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So, if I differentiate xe to the x minus 
e to the x, I don't need to add the plus 

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C because if I differentiate a constant, 
I just get 0. 

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All right, this is a derivative of a 
difference so it's the difference of the 

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derivatives. 
But now this is a derivative of a 

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product, so I'm going to use the product 
rule. 

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I'm going to take the derivative of the 
first, so the derivative of x, times the 

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second, plus the first times the 
derivative of a second, right? 

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That's the product rule. 
And then I subtract, well, the derivative 

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of e to the x is just itself. 
And I've got the derivative of x, which 

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is 1 times e to the x plus x times the 
derivative of e to the x e to the x minus 

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e to the x. 
And here's the slightly exciting part, 

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right? 
This and this cancel, and all I'm left 

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with is just x times e to the x. 
So, in fact, we have found an 

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anti-derivative for x e to the x. 
Here it is. 

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And of course, that makes sense because 
integration by parts is just a product 

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rule in reverse. 
Now, we can use the same trick to attack 

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similar integration problems. 
For example, let's say you want to 

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anti-differentiate some polynomial in x 
times e to te x. 

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You could do this with parts. 
Well, how? 

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Well, I've made this be u and I make this 
be dv. 

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And why is that such a great choice? 
Well, think about what parts lets you do. 

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Parts lets you differentiate part of the 
integrand if you're willing to 

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anti-differentiate the rest. 
But anti-differentiating e to the x is 

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paying no price at all because it's its 
own anti-derivative. 

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And, if you differentiate the polynomial, 
then you reduce its degree. 

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So, by doing parts enough times, 
eventually you're just 

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anti-differentiating e to the x by 
itself, which you can definitely do.