 Hello and welcome to this screencast on integration by parts. Here's our function and I've already set up a reminder of the formula for integration by parts. So this time we're going to integrate 2x e to the x with respect to x. So the first thing to notice is this isn't an elementary antiderivative. We don't just know it right off the bat. And it's also not a composite function, so we can't use substitution. So integration by parts is a good choice in this case. Another reason that it's a good choice is that over here in the formula for integration by parts we have two things multiplied together, u times dv. And in our formula over here we have two things, two different functions multiplied together, a 2x and an e to the x. So this is probably how we're going to break it up. The only question is which of these is the u and which of these is the dv? Well, a general rule I have for figuring out how to choose u and how to choose dv is that we shouldn't make the resulting integral any worse than when we started. So that means whatever you pick for u should have a simple derivative and it shouldn't be any worse than what you start with. And whatever you pick for dv should have a simple integral and it certainly shouldn't be any worse than what you start with. So in that case in terms of integrating I'm going to pick e to the x to be by dv. And the reason I'm picking that is that when I integrate it I'm only going to get e to the x again. If I had chosen 2x my integral would be x squared and that is bigger and probably less easy to work with. So that leaves only 2x to work with. And here I remember something that's easy to forget. The dx goes with the dv. So the derivative gets the dx on the end of it. Alright, so now starting with the u I'm going to calculate du. That's the derivative of u which is just 2 dx. Don't forget the dx. And now starting with the dv I'm going to calculate v. That's the integral which is still just e to the x. Alright, so using this I'm going to follow the formula for the integral of u dv. So I already have something in the form udv and this is going to become using the integration by parts formula u times v minus the integral of v du. Don't forget that dx in there. Okay, and that's it. The integration by parts formula has transformed this from an integral that we don't know how to do into one that's not too bad. So I'm going to simplify this a little. I get a 2x e to the x minus 2 times the integral e to the x dx. And that integral of e to the x with respect to x is something we know how to do. So I get 2x e to the x minus 2 times e to the x and here is where the plus c comes in. Because now that I've taken the antiderivative I need the general antiderivative. And this is my general antiderivative for 2x e to the x. It's 2x e to the x again minus 2e to the x plus a constant. So let's check this. So here on the next slide I've got what we started with. I can always check by taking a derivative. So to make sure I have an antiderivative I'm going to take the derivative with respect to x of 2x e to the x minus 2e to the x plus a constant. That first derivative is a product rule. This happens a lot because integration by parts is the product rule done backwards. So for the product rule that first part is the derivative of the first part times the second plus the first part times the derivative of the second. Now I get to the minus 2e to the x whose derivative is minus 2e to the x and the constant becomes a zero. These two parts cancel nicely and I'm left with just 2x e to the x which is my original integrand. So this is a check that I have an antiderivative.