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Evaluate the given integral.$$\int \frac{3 x+2}{\left(3 x^{2}+4 x+1\right)^{5}} d x$$

$$\frac{-1}{8\left(3 x^{3 / 2}+4 x+1\right)^{4}}+c$$

Calculus 1 / AB

Chapter 5

Integration and its Applications

Section 3

The Substitution Method

Integrals

Baylor University

University of Nottingham

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 integral.…

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Evaluate the given integra…

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if we take you to be this inside function and the denominator, which is three X squared plus four X plus one. Then when we take do you we get, do you use equal to six x plus four d X, and at this step, we can divide through by two. To get do you over to is equal to three x plus two DX, and that allows us to remove this three x Plus two DX and rewrite this integral as one half times the integral of you to the minus fifth do you, which is equal to one half times one over minus five plus one times you to the minus five plus one plus C, which is equal to 1/2 times minus 1/4 you to the minus four plus C, which is equal to minus 1/8 times three X squared plus four x plus one to the minus four plus C, and that completes the problem.

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