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Evaluate the integral by making the given substitution.

$ \displaystyle \int x^2 \sqrt{x^3 + 1} \, dx $, $ u = x^3 + 1 $

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$$\frac{2}{9}\left(x^{3}+1\right)^{3 / 2}+C$$

01:16

Frank Lin

Calculus 1 / AB

Chapter 5

Integrals

Section 5

The Substitution Rule

Integration

University of Michigan - Ann Arbor

University of Nottingham

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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Okay, We've been told that you is x cubed plus one, which means I'm gonna take the derivative we get D'You is three x squared DX which means d axe is do you divide it by three X square? Okay, Now that we have this, we know that we can integrate and we know we have integral X squared times Spurt of x cubed plus three. So use the power rule which is increase the exploding by Juan divide by the new experiment. When you're integrating, it's the power rule is in a textbook. This gives us 2/9 times square root of X cubed plus three huge plus c

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