 In this video, we're gonna discuss the solution for question 10 on the practice midterm exam for calculus two math 12-20. And we're asked to compute the anti-derivative of x e to the negative x dx. Now this one right here when I look at it, this one screams integration by parts, right? Where we're gonna set u equal to x and dv equal to e to the negative x dx. And so why does this one scream integration by parts? Well, let's first remember what integration part says, right? If you're integrating u dv, this week will u v minus the integral of v du. So we have to look for a function for which we want to take its derivative. And we wanna look for a function for which we wanna take its anti-derivative. Well, because of the e to the negative x, the anti-derivative and the derivative are really just basically the same thing. It'll be e to negative x. There's not really much of a difference there, but there is an advantage by taking the derivative x, it's gonna vanish, right? The derivative of u will just be dx there. It just disappears. And then like I said, the anti-derivative of e to negative x is negative e to negative x. And so if we apply that principle, our integral will look like we have to take v times u, which gives us negative x e to negative x. And then we have to subtract the integral of negative e to negative x dx. For which case we have a double negative, they cancel out. And now I just have to find the anti-derivative e to negative x and look at that, I've already done it. How wonderful. And so our final answer would then be negative x e to negative x. We're then gonna negative e to negative x. And perhaps the most important part here is remember the plus c, right? This is an indefinite integral. We're looking for the family of anti-derivatives. And so forgetting the c is as if we take the initial value that our c was zero, right? We don't want that. So for full credit, we do need the plus c. You don't wanna forfeit that point there. But otherwise we get this very nice integration by parts. You're gonna wanna make sure you don't integration by parts for this exam.