 So, let's look at integration by parts. The secret that not everybody is going to tell you is the following. There's actually only two real integration techniques. And the first one is U-substitutions, which we saw earlier. And the only other integration technique is this integration by parts. Everything else, all other integration techniques are based on algebraic and trigonometric identities. Everything else you do as a quote-unquote integration technique is really algebra, or sometimes it's trigonometry. And to be sure, these things are then combined with either U-substitutions or integration by parts. So, let's focus on what integration by parts is. So, let U of V be two functions. When we looked at switching the direction of integration, we made use of a useful geometric relationship. Informally, the way we might look at this relationship is if I want to find the definite integral, if I want to find the area of a region, that's going to be the integral of U dV. And what I looked at was the fact that if I piece this together with another region, which in this case is going to be corresponding to the integral of V dU, we get that by looking at the representative rectangles. If I put those two together, I get a rectangle. And I might describe the area of the rectangle as being U times V. And this is an informal argument, and we're doing a lot of hand waving here, but it does suggest a more general relationship, and it turns out that this relationship allows us to take one integral, which we might or might not know how to do, and replace it with a different integral, and then some other expression here. And so the idea is that if I have an integral that I can't do, maybe I'll replace it with a different integral, and possibly I might be able to do that second integral. So, for example, let's take a look at the finding the anti-derivative of x e to the x dx. If we lived in a good, kind, and caring universe, then every problem would indicate exactly how it's supposed to be solved. We don't live in that universe. So, what that means is that in general, when we run into a problem, we don't know how we're supposed to solve it, and so the general strategy here that's useful is always try the easy things first, because if they work, you want to spend a lot of time, and if they don't work, you want to waste it a lot of time. So, first thing we might do, we might actually recognize this as the integrand, as the derivative of some function. Maybe we recognize it, and we could write down what the anti-derivative is without any difficulties. Well, probably that's not going to be the case, so maybe the next thing we'll try, again, u-substitution is probably the easiest of the possibilities. Can I solve this using a u-substitution? And we have to try these things before we know for certain, but in this particular case, if we do that, we find we can't. So, if I apply integration by parts, that's my next third choice. I need to split the integrand into two parts. I need a u-part and a dv-part. And the key difference between the two is the u-part is going to be differentiated. The dv-part is going to be anti-differentiated. So, maybe I'll make some choices. I'll let u be x, and dv is all the rest of it. And I need to differentiate the u-part, and I need to anti-differentiate the dv-part. And let's see what happens there. So, I now have my terms, u-v-d-u-d-v. And I can substitute these back into my relationship for integration by parts. So, the first hundred or so times you do that, it's probably worth writing down what that relationship is, integral u-d-v, is u-v minus integral v-d-u. And I can substitute things in right from this statement. I have u is equal to x, there's my u. I need my dv-d-v, e to the x dx, so there's my original integrand. Let's see, I need v, that's going to be e to the x, it appears here and here. And then finally my du is going to be dx. And I have a new expression. Now, remember the difficulty here was we didn't know how to handle the left-hand side. And so the challenge is, can we handle the right-hand side? And important to remember, integration by parts does not actually give us an anti-derivative, but it changes our problem. And hopefully we can have an anti-derivative that we can, in fact, determine. And in this particular case, it is something that we can find. This is the world's easiest anti-derivative, is e to the x. Don't forget that constant of integration, because we are still dealing with an indefinite integral or the anti-derivative.