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Evaluate the given integral.$$\int\left(2 x^{3}+3\right) 6 x^{2} d x$$

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

Calculus 1 / AB

Chapter 5

Integration and its Applications

Section 3

The Substitution Method

Integrals

Missouri State 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.

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Evaluate the integral.

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Evaluate integral

we have the integral of to execute plus three times six x squared. And instead of distributing the six x squared weaken, solve this using a use substitution notice that the derivative of To Execute Plus three is six x squared, so we can use that information to rewrite this integral. So let's take you to be two x cubed plus three, and we get that, do you DX that's equal to six x squared. And this allows us to rewrite this integral in terms of you because D. U is equal to six x square DX. So this becomes the integral of you times. Do you? And what we've just done is replace six x square D x. Let's use a different color. 666 times X squared DX was replaced with do you and to execute plus three that was replaced by you. So we get that this is equal to one half times you squared plus c. And now we have to go back and replace you writing that in terms of X. So this is one half times to execute plus three squared plus C, and that completes the problem

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