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Evaluate the given integral.$$\int x^{2} e^{-2 x} d x$$

$$-\frac{1}{4}\left(2 x^{2}+2 x+1\right) e^{-2 x}+c$$

Calculus 1 / AB

Chapter 5

Integration and its Applications

Section 9

Two Integration Techniques

Integrals

Campbell University

University of Michigan - Ann Arbor

Idaho State University

Boston College

Lectures

05:53

In mathematics, an indefinite integral is an integral whose integrand is not known in terms of elementary functions. An indefinite integral is usually encountered when integrating functions that are not elementary functions themselves.

40:35

In mathematics, integration is one of the two main operations of calculus, with its inverse operation, differentiation, being the other. Given a function of a real variable (often called "the integrand"), an antiderivative is a function whose derivative is the given function. The area under a real-valued function of a real variable is the integral of the function, provided it is defined on a closed interval around a given point. It is a basic result of calculus that an antiderivative always exists, and is equal to the original function evaluated at the upper limit of integration.

00:56

Evaluate the integral.…

02:02

01:35

Okay, So we want Teoh, um, find their family integral. So let's use our tabular integration Brightpoint. So we're gonna have repeated integration or different ation. It's got this, and this year is going to be repeated integration. So we're gonna differentiate x word. So this is going to be two X and then we're gonna have to and then zero and then for integration. Well, this is e to the power of start. It should be a negative two x. Okay, So we started with you depart of native to X, and then it's differentiate this we're gonna get Okay, so this is going to be negative. One half, you get a part of negative two X and then we're gonna have one over full of you to depart. Made of two X, Been in negative 1/8. It is two times war. You department, you have to. Thanks. So, no, it's drawn out are falling arrows. And here we're gonna have plus minus and plus So if you are following, integral is equal to X squared times. Negative one half and then we have eating Harvick two X and then we have Let's see, we have plus two x times the negative of the folowing. So this is going to be minus to over four. That's one half. So that's X over two, and then we have need to depart Negative two X and then plus to over eight. That's going to be a negative 1/4 on each of power of negative two x I didn't have plus C.

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