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Evaluate the given integral.$$\int \frac{5^{3 x}}{5^{3 x}+2} d x$$

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

Calculus 1 / AB

Chapter 5

Integration and its Applications

Section 3

The Substitution Method

Integrals

Missouri State University

Oregon State University

Harvey Mudd College

University of Michigan - Ann Arbor

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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03:45

if we take you to be five to the three x plus two than D u D E X. That's equal to three times Ellen of five times five to the three x by chain room. And now we can replace five to the three x dx by do you over three times the natural log of five. So this becomes 1/3 times Ellen of five times the inter groom of One Over You, which is equal to 1/3. Ellen of five times Ellen of five to the three x plus two plus C. And we don't have to worry about the absolute values because five to the three X is positive. It's greater than zero, and that completes the problem.

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