 In this video we provide the solution question number five from practice exam number one for math 1220, and we have to compute the indefinite integral x times e to the negative x dx. This is one where we want to use integration by parts. Notice that if we take the derivative of x it'll vanish. When you take the derivative or the anti derivative of e to the negative x it really makes no bit of a difference whatsoever. They're actually the same thing. So I'm going to use integration by parts. I'm going to set x equal to u. Therefore du will equal dx. I'm going to set dv equal to e to the negative x dx. Therefore v is going to equal negative e to the negative x like so. And so then using the formula for integration by parts we're going to get a u times v. So that's a negative x times e to the negative x. We then subtract from that integral v du so we get negative e to the negative x dx like so. Then we're going to evaluate this integral. It is a double negative. So you get negative x e to the negative x plus the integral e to the negative x dx. But this integral we've already computed the anti derivative of. It's just going to be negative e to the negative x. So we end up with negative x e to the negative x minus e to the negative x plus a constant. And then we should write this on the line negative x e to the negative x minus e to the negative x plus a constant. And that is then the correct anti derivative.