 Hello, and welcome to the screencast, a definite integral using integration by parts. Today, we're going to take a look at this integral, which is the same integral as 1 from example 2 in this section. However, this time it's a definite integral. We have the limits of integration from 0 to 1 on this. So right now, we're not going to focus a whole lot on the integration by parts part of this. So pause the video for a moment and make sure you know how you would choose u and dv and how you'd use integration by parts to write out the anti-derivative. All right, we're back. I would choose u to be 2x and dv to be e to the x dx, which leaves v to be e to the x, and du to be 2 dx. Now here's where the interesting part starts. We're going to use integration by parts as usual, and we're just going to mark down that we need to evaluate the result from 0 to 1. So we're going to remind ourselves that we need to do a definite integral. But we're not going to actually evaluate it until the last step. So here's how we're going to do that. The first step of integration by parts is to write out u times v. So that's 2x e to the x. And I'm going to immediately write next to it this bar with the same limits of integration, 0 to 1. This means evaluate this function from 0 up to 1. Now I'm going to write the next part of integration by parts minus the integral, and it's going to be the definite integral, of v du. And that's e to the x times 2 dx. So this is exactly the same as what we would do for integration by parts, u times v minus the integral of v du. But we've left the integration limits here on the part out front, as well as here on the resulting integral. And both of those are necessary because we're going to have to evaluate everything that we get out of this integral. All right, let's simplify, but I'm not going to evaluate anything yet. I'm going to keep those bars out front until everything else has been evaluated. So I'm going to evaluate this integral, which I happen to know in antiderivative 4, 2e to the x. And I'm going to write the bar again, evaluate it from 0 up to 1. Just like in calc 1, it's a good idea not to do all of the integration in your head and all in one step. Instead, these bars can help remind you that you need to do integration by parts first, and then evaluate the definite integral second. All right, now that we've got all the integrals taken care of, it's time to actually evaluate these. So just like any definite integral, I'm going to substitute the top value, 2 times 1 times e to the 1, minus, and I'll substitute the bottom value, 2 times 0 times e to the 0. Now this all corresponded to just this first part right here. And now for the second part, that's going to go over here a little ways, and I'm going to evaluate 2e to the x from 0 to 1 as well. So that's 2e to the 1 minus 2e to the 0. So notice how I evaluated each of these parts from 0 to 1. I had subtraction in the middle. And I kept track of my parentheses carefully, because all this subtraction needs to be done for entire terms at a time. With a little bit of simplification, this turns into 2e minus 0, and then distributing the minus sign carefully, minus 2e plus 2, because e to the 0 is 1. The 2e's cancel out, and all that I'm left with is 2. So something to notice here is I have a number in the end. As I should, this is a definite integral. It represents the net signed area between the x-axis and 2x e to the x over this interval from 0 to 1. And so I should have a number. If you end up with variables, it's likely because you forgot this very first step that we took, where I evaluated 2x e to the x from 0 to 1 right here. If you forget to write that bar from 0 to 1, then you may end up with variables in your answer. And if you did, go back and double check and make sure you've evaluated every part of this integral.