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Evaluate the given integral.$$\int \frac{e^{2 x}}{\left(e^{2 x}+1\right)^{3}} d x$$

$$\frac{\ln \left(5^{3 x}+2\right)}{3 \ln 5}+c$$

Calculus 1 / AB

Chapter 5

Integration and its Applications

Section 3

The Substitution Method

Integrals

Oregon State University

Harvey Mudd College

University of Michigan - Ann Arbor

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.

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Evaluate the given integra…

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Evaluate the integral.…

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Evaluate the integral.

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Evaluate the definite inte…

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evaluate Integral

in the denominator of this function is e to the two x plus one cubed. So let's replace each of the two x plus one by you. That way do you is equal to two times each of the two x dx. And now we can, um, get rid of each of the two x dx, replacing that by D you over to and rewriting this integral as 1/2 times the integral of you to the minus three, which is equal to 1/2 times you to the minus three plus one all over minus three plus one plus C, which is equal to 1/2 times minus 1/2. And you is each the two x plus one all raised to the minus two plus C, which is equal to minus 1/4 times E to the two x plus one to the minus two plus c, and that completes the problem.

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