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Evaluate the integral.

$ \displaystyle \int \sqrt{x} e^{\sqrt{x}}\ dx $

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$$2\left(y^{2}-2 y+2\right) e^{y}+C=2(x-2 \sqrt{x}+2) e^{\sqrt{x}}+C$$

Calculus 2 / BC

Chapter 7

Techniques of Integration

Section 5

Strategy for Integration

Integration Techniques

Oregon State University

Harvey Mudd College

University of Michigan - Ann Arbor

Lectures

01:53

In mathematics, integration is one of the two main operations in calculus, with its inverse, differentiation, being the other. Given a function of a real variable, an antiderivative, integral, or integrand is the function's derivative, with respect to the variable of interest. The integrals of a function are the components of its antiderivative. The definite integral of a function from a to b is the area of the region in the xy-plane that lies between the graph of the function and the x-axis, above the x-axis, or below the x-axis. The indefinite integral of a function is an antiderivative of the function, and can be used to find the original function when given the derivative. The definite integral of a function is a single-valued function on a given interval. It can be computed by evaluating the definite integral of a function at every x in the domain of the function, then adding the results together.

27:53

In mathematics, a technique is a method or formula for solving a problem. Techniques are often used in mathematics, physics, economics, and computer science.

02:28

Evaluate the integral.…

01:13

Evaluate the indicated int…

01:33

04:15

00:57

00:32

Evaluate the indefinite in…

01:12

04:51

06:00

integral

Let's start off with the use of here. Let's take you to be the square root of X so that do you is one over to rue x t x, and we could write This is one overthrew you d x and then go ahead and solve this equation here for DX, sir, T X equals to you, do you? So we can rips rewrite this integral as well. We see a radical X there, That's just you. And then here we see into the U and then the ex. Well, we just found that that's just who you deal. Swish. Go ahead and pull out the two there. Use clear each of you. And unless you have some formula memorized for something like this, we could try integration by parts. So here, let's take w two bu square D w is to you to you. And then TV is either the huge deal, so v equals either the u and recall the formula here. So this will be w V minus inner growth. BTW. So I'm just gonna pull out the two here, and then I have w times V. So that's just you square either the U and then my tous in a girl And then I have VD you v d w Excuse me. So we go ahead and pull out this extra to here and then inside the inner girl, we just have you and that either the you do you. So let's go ahead and close the front to see there. So this is our latest expression down here. Let me go ahead to the next page and rewrite this. No. Okay, so then now we have a very simple looking in the world. Someone we just did the one before had its two up there, but ours does not. So let's get rid of that too. So since it's so similar, let's go ahead and try another integration by ours here. I'm just using it. If you want, you could use their usual others here of a sense I've used up you, V and w Let me bring in C here to see you. D Z equals do you And then here means another letter DT equals either the u D'Oh. As for this over here, so t equals interview and then using these letters integration, my purse will be so let's go ahead and write that up. So the first part we don't have the degree and then minus two and then now we have Z times t so that you need to the U and then minus integral and then he and times easy solution. Either the you do you. And we know this intern girl. So let's just go ahead. And also most of plant these twos. So to you, swear you and then minus and that here will have a four. You do the U and then watch out for that double minus will get a plus and then we have two times two and then in overall, this is just eating you and go ahead and add that constant of integration. CNN's now the last step here Messages used original substitution to go ahead and rewrite everything back in terms of X. So the first term here. That's too, you square. So that's two accident seeds in the Radical X minus four radical ex erotica Lex and then for erotic alexe. Keep batting that constant of integration, and there's a final answer

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