 Hello, and welcome to the final screen cast about substitution. And this time we're going to do one with a definite integral. Okay, so let's look at our example. We've got the integral from negative 1 to 5. Remember, you always start at the bottom and go to the top of e to the 2x dx. Now, this particular function may look similar, hopefully, to something that you've done before. This was an earlier video, but I just added endpoints on it. Okay, so you've got a couple of options when you go to do this. So we'll pick our substitution, and then we're going to change our endpoints, and then we're going to finish the problem. Okay, as we're kind of going through it, I'll give you different ways that we can do this one. So again, take a second, think about what would you pick for your U, and why does your substitution even make sense with this particular problem? Right, well, it makes sense because this is a composite, right? There's an inside function, there's an outside function. If I were doing the derivative of this particular function, I would be using the chain rule. So that's how I know that I want to do U substitution. Okay, so my inside function with this one is 2x. I do my derivative, and I get just a 2 dx. Okay, we're matching again, so I just see just a plain old dx in the integral. So I have to multiply by a half. So half du is going to give me that dx by itself. Okay, so this is where, like I said, you've got some choices. If you want to just do the integral in terms of U, then, which is what I'm going to do, we are going to change these endpoints then. Because remember, these endpoints are technically x's. So let me rewrite this original integral. So this was x equals negative one, and x equals positive five, e to the 2x dx. Now, we typically don't do this because first of all, it looks ugly. And second of all, if there's an x out here, then we know everything's in terms of x, but this is just a good way to remind ourselves of that. Okay, so if x is going to be negative one, if I plug that into my U substitution here, what's our U going to be? Well, U's going to be 2 times negative one, which is going to give us negative two. If our x is five, plug that into our substitution. So U's going to be 2 times 5, which is 10. Okay, so now I'm going to go ahead and rewrite my whole integrals in terms of U's. So first, let's write out those endpoints that we just changed. So U is now negative two, and U is a positive 10. My E function didn't change because that was my outside function. But my inside function here, let me highlight that in a different color. This function up here in the exponent, my 2x, is my U. So I can go ahead and rewrite that. And then our dx, we said, let's see, going back here is one-half du. Okay, so just as we've done before, we can go ahead and pull that one-half out front of the integral to make it look a little bit nicer. We now know everything's in terms of a U, so I'm just going to write negative 2 to 10, e to the u, du. Okay, so now you need to do the antiderivative. So can you think of a function whose derivative will give you e to the u? And this one's almost so easy, it's challenging. That's just e to the u, the one-half comes along. And now this is a nice part about changing your endpoints over to U's. It's now I can just evaluate it from negative 2 to 10 and not worry about plugging my u back in, okay? Because remember, these endpoints are in terms of u, not x. So let's say you didn't change your endpoints and you just kind of solve this as an indefinite integral. Then you'd have to plug your u back in and then plug in your accent points, okay? So don't confuse the two methods, just pick one and stick with it. So like I said, for this one, these are our u's, so we could just plug those back in. So we have one-half e to the 10th minus one-half e to the negative 2, okay? It's kind of ugly because they're e's and I'm not going to change these to decimal, so I'm just leaving them as e's. But if you want to run this in your calculator, I'm sure it'll give you some kind of an approximation for that. That you can then check in the area under the curve or whatever you want to do there. All right, thank you for watching.