 Hello, and welcome to a screencast about integration by parts. So the example we're going to be looking at today is evaluating the indefinite integral of x times the sine of x dx. And just to remind ourselves, up here is the integration by parts formula, and that says if you're going to do the integral of u dv, that's going to be equal to u times v minus the integral of v times du. And it's probably a good idea to kind of understand where that formula comes from, rather than just memorize it. So talk to your professor or somebody else, and hopefully they can help you figure that one out. So anyway, you want to use integration by parts whenever you think of the product rule. You know, so if I were doing the derivative of x times the sine of x, I said the word times in there, so I know I'd have to use the product rule. Also doing a u substitution on this one wouldn't help at all, right? I mean, you can kind of have an inside function, but that's just x, so that's not much of a function. You go to do the derivative of it, it's really not going to give you much. So anyway, like I said, when you know you say that we're times, or when you think of the product rule, that's always a good idea to know, basically, that you're going to be using integration by parts. Okay, now, next choice is we're going to have to figure out what's going to be our u, and what's going to be our dv. And then we're going to want to do the derivative of u, because we've got a du part in here, and then we're going to want to integrate dv, because we've got a v part in here. So again, remember going this direction, we're going to differentiate, but if we're going this direction, we're going to be integrating. Okay? So basically, you want to pick a dv so that it's easy to integrate. Also, you're going to want your resulting integral to be easier, not bigger. So sometimes, like for this one, the sine of x, that function's easy to integrate and differentiate. So that one kind of is a crapshoot, but then for picking the u, you usually want it to make something smaller. So if I were to let u be sine in this particular example, that just goes to cosine. Well, that just kind of rotates back and forth, so that doesn't really get smaller. So in this case, I'm going to let my u be x, because when I do the derivative of x, that's definitely going to get a lot simpler, a lot smaller. So that means that my dv in this case is going to be the sine of x, dx. Okay? So as my arrows show, when I differentiate, differentiate x, I get dx. I integrate sine of x, that's going to give me negative, oops, cosine x. My brain's working faster than my hand today. Negative cosine of x. Okay? So now I got my pieces in here. Now I can go ahead and put those into my integration by parts. So I've got my integral of x, sine x dx, is equal to u times v, so that's going to be x times negative cosine x, and I'll pretty that up in a second. Then I have minus my integral of v again, so that's negative cosine x, d u, which in this case is just dx. Okay? So is my next integral easier? You better believe it, right? I don't have a product anymore, and I certainly know how to integrate negative cosine of x. Even better, I'm going to go ahead and move that negative on the outside of this integral. I'm also going to rewrite this first piece. I'm going to go ahead and pull that negative out front of this first term. And then because I had a negative in my integration by parts formula and a negative in my integral, that gives me a positive, cosine x dx. Cosine's a function I can easily integrate, so my final answer becomes negative x cosine x plus the sine of x, of course, plus c, because we want to make sure we generalize these since it's an indefinite integral, and that's our answer. Okay? Now, let's say I had not chosen, you know, wisely, maybe it's a good way to put it, then maybe I kept getting more complicated integral. Well, then you want to stop, you want to start over. Maybe switch your U and D V around and try again. Okay? So doing these problems, it's always good to start them on scratch paper, especially if you're going to be turning something into somebody. Start it on scratch paper, you know, try it out, but you've got to not be willing to give up. So that's, I think, probably my best piece of advice with these problems. All right? Thank you for watching.