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Evaluate the indefinite integral. Illustrate, and check that your answer is reasonable, by graphing both the function and its antiderivative (take $ C = 0 $).

$ \displaystyle \int xe^{-2x} dx $

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$-\frac{1}{2} x e^{-2 x}-\frac{1}{4} e^{-2 x}+C$

Calculus 2 / BC

Chapter 7

Techniques of Integration

Section 1

Integration by Parts

Integration Techniques

Harvey Mudd College

University of Nottingham

Idaho State University

Boston College

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.

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Evaluate the indefinite in…

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The problem is, you want it. It's a definite integral illustrated and check that your answer is reasonable. Graphing boasts of function and his anti derivative integral axe temps into a negative too. Thanks, Jax. With this problem, we can use integration parts to you. Violet, Integral. Homer is into girl off you, Thompson from yaks. It's cultural. New tats, wing minus. Into a girl, you probably have to weigh the axe. Our problem? We can let you asleep to ACS and the prom is about you. You too, Nick. To two packs. Then you promise you goto one being secret. Negative one half intro ninety two packs. Now, by using this formula to censure girl is true Newtown Sweeney. So this is negative. One half have sex house into negative two acts, then minus integral. You prom times. We This is a class one. Half need to negative. Too lax. Yes. Now this is They come to negative half extensively. Too negative. Too lax. Minus one fourth into negative to x and plus cast member si. Now, with you, some grafting tours to graph, uh, picture off function Y is equal to x times connected to ACS and its anti terror votive. I see little clock, Thiss says. Brew on is anti derogative off this function. Now I can see while you off a blind one. At some point. X is the slope off the hand in line helps up blue curve. At this point, these degenerative off blue one is right. Now that's low cat, the slope of the tangent line. Off the bloke, one X goes from negative infinity to infinity slope of attended line. It's a little curve. It was from a bacterium vanity and to Teo onto part Tio. Some pause to number and teo zero. Now let's look at why do you have my teacher? My axe goes from negative infinity to infinity, the wily off. They're like cave escorts from because from neck to infinity to zero, some passed to number and two zero, since the other sets the same while you have the right kind of the same as the slope of a tan in the line after Blocher. So our results is read some of reasonable

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