 We can also apply integration by parts, even if we don't know what the function is, as long as we know some information about it. Since the integral of u dv is equal to uv minus the integral of v du, then if we know the values of u and v at the endpoints, and the value of one of the integrals, we can find the other. So for example, suppose I know that f of 0 and f of 10, and the value of some integral, let's find the integral of xf prime of x. So we'll set up our integration by parts. We'll choose a u and a dv. That means we need something to differentiate and something to anti-differentiate. And let's try u equals x and dv equals f prime of x. So we'll differentiate and anti-differentiate. And so the integral of xf prime of x, well that's uv minus the integral of v du. And so we need to evaluate xf of x at 10 and at 0. And we need to know the value of the integral of f of x over the interval 0 to 10. But how can we possibly find that when we don't know what f of x is? If only there was some way of knowing the value of this integral. What can we do? What can we do? Oh wait, right here. And we need the value of f of 10 and f of 0, well it's being multiplied by 0 so we really don't care. But the values are going to be... So for more complicated problems suppose we have this information about the integral and certain values at the end points. So again we need something to differentiate and something to anti-differentiate. So let's pick these. Differentiating and anti-differentiating. And the problem here is we don't know anything about this anti-derivative of f of x. So let's choose u and dv in the other way and find the derivative and the anti-derivative. And so integration by part says that this integral is uv minus the integral of v du. We can evaluate the first function at 5 and at 0. The integral can be cleaned up a little bit and we know the value of the integral. And the important thing to remember is that all of our rules of differentiation and integration still apply. So suppose I have a ton of information. Since we know something when the integrand includes the derivative of f of x we choose u to be f of x so we can differentiate it and dv to be the rest of it. Which will anti-differentiate. Applying integration by parts and evaluating. We need to evaluate this mass at 3 and at negative 1. Now for this mass well we know something about the integral of f prime xf prime and x squared f prime. If only there was some way we could break this integral apart. Oh wait the integral of a sum is the sum of the integrals and so we could rewrite this. And the constant multiplier rule allows us to rewrite this further. And we know the values of these integrals and so we compute the value of the integral we want.