[MUSIC].
The basic idea of integration by parts is that it lets you differentiate part of
the integrand, but only if you're willing to pay a price.
And that price is anti-differentiating the other part of the integrand.
Well, let's try it. For example, let's find an
anti-derivative of x times e to the x. If we're going to attack this integrand
by parts, then I've got to pick a u and a dv.
I'd be willing to differentiate x. That'll make x just go away.
And the price I'd have to pay is to anti-differentiate what remains.
But anti-differentiating e to the x is not much of a price to pay because e to
the x is its own anti-derivative. So let's set that down.
Let's set u to the x, because I'd like to differentiate that part.
And I'm willing to anti-differentiate what remains.
Now, let's figure out du and v. Now if u is x, then du is dx.
And then, I've got to pick an anti-derivative for dv.
Now, in principle, there's a ton of anti-derivatives I could pick, right?
e to the x plus 17 differentiates the e to the x.
But I'll just pick the nice one. I'll pick e to the x.
Now I've got u, dv, v, and du. We can put it all together.
Parts tells me that an anti-derivative u dv is uv minus anti-derivative v du.
So, in this case, the anti-derivative of u dv is u times v, x times e to the x
minus anti-derivative of v du, which is just dx.
But, I know how to integrate e to the x. Anti-derivative of e to the x is just
itself. So, the anti-derivative of x, e to the x
is x, e to the x minus e to the x plus some constant.
We did it, and we can check our answer. So we have to differentiate this and make
sure I get x, e to the x. Let's try it.
So, if I differentiate xe to the x minus e to the x, I don't need to add the plus
C because if I differentiate a constant, I just get 0.
All right, this is a derivative of a difference so it's the difference of the
derivatives. But now this is a derivative of a
product, so I'm going to use the product rule.
I'm going to take the derivative of the first, so the derivative of x, times the
second, plus the first times the derivative of a second, right?
That's the product rule. And then I subtract, well, the derivative
of e to the x is just itself. And I've got the derivative of x, which
is 1 times e to the x plus x times the derivative of e to the x e to the x minus
e to the x. And here's the slightly exciting part,
right? This and this cancel, and all I'm left
with is just x times e to the x. So, in fact, we have found an
anti-derivative for x e to the x. Here it is.
And of course, that makes sense because integration by parts is just a product
rule in reverse. Now, we can use the same trick to attack
similar integration problems. For example, let's say you want to
anti-differentiate some polynomial in x times e to te x.
You could do this with parts. Well, how?
Well, I've made this be u and I make this be dv.
And why is that such a great choice? Well, think about what parts lets you do.
Parts lets you differentiate part of the integrand if you're willing to
anti-differentiate the rest. But anti-differentiating e to the x is
paying no price at all because it's its own anti-derivative.
And, if you differentiate the polynomial, then you reduce its degree.
So, by doing parts enough times, eventually you're just
anti-differentiating e to the x by itself, which you can definitely do.