 Hello, and welcome to the screencast on integration by parts. In this example, we'll be taking a look at the integral x squared natural log of x with respect to x. So this is not necessarily an easy choice when figuring out how to use integration by parts. The reason is we have two things multiplied together, an x squared and a log x. But it's not really obvious which way to choose u and which way to choose dv. So in a moment, I'm going to ask you to pause the screencast and choose your own u and dv. And then you can start again and see if you agree with what I choose. As advice, remember that you're going to choose u so that you can do the derivative du. And you're going to choose dv so that you can do its integral v. And you in general want to choose your parts so that the resulting integral is no worse than what you started with. So pause right now, write down your own choice, and then we can come back and see what I got. All right, we're back. My choice is to have u be the natural log of x and dv to be x squared dx. Why would I do this? Well, if I chose the other way so that I had natural log be dv, I'd have to integrate it. And we don't know the integral of natural log of x. That means I'm pretty much stuck making u be natural log of x since I do know its derivative. Now, dv being x squared does cause a little bit of a problem because it makes v, which is 1 third x cubed, more complicated than what I started with. But that's the price we pay for having to be able to calculate du, which is just 1 over x dx. And that's not too bad. So let's give this a try. Sometimes with integration by parts, you do have to try one or more times, or you do have to use another technique when you're done. And this is a good example of how persistence can really pay off in calculus. So we're going to try this, see if it works. And if it doesn't, we'll try again. And that's OK. So I'll rewrite my integral. And then using integration by parts, I have u times v. So that's natural log of x times 1 third x cubed minus the integral of v du. So there's v and here's du. And now that I've written this out, this actually looks like it's going to turn out fairly well. So I'll recopy the first part minus. And after I move that 1 third out front, I can simplify this. So I can see right here that I've got an x and an x cubed. And that is going to simplify down to just x squared dx. And that's good news. It turns out my choice here worked out well because the derivative of natural log cancels with part of the integral of x squared. So here rewriting again, I can do an anti-derivative for x squared. That's just another 1 third x cubed plus an arbitrary constant. And at this point, I could simplify. But I'm effectively done with the anti-derivative. And as always, you can double check by taking a derivative and simplifying and seeing if you end up with what we started with, x squared natural log of x. So one lesson from this is sometimes you have to focus on the dv part and make sure that you choose something for dv that you can actually integrate. And occasionally, that forces your hand and you have to choose things in a certain order as we did here. Another lesson from this is that sometimes you will make mistakes. If you chose differently, that's OK. Try writing out the integral. See if you can do the result. And if you can't, go back, use a new piece of paper, and try again. Integration often works out like that. And persistence pays off.