   Chapter 5, Problem 42RE ### Single Variable Calculus: Early Tr...

8th Edition
James Stewart
ISBN: 9781305270343

#### Solutions

Chapter
Section ### Single Variable Calculus: Early Tr...

8th Edition
James Stewart
ISBN: 9781305270343
Textbook Problem

# Evaluate the indefinite integral. Illustrate and check that your answer is reasonable by graphing both the function and its antiderivative (take C = 0). ∫ x 2 x 2 + 1 d x

To determine

The value of the indefinite integral function x3x2+1dx.

Explanation

Given information:

The integral function is x3x2+1dx.

Consider (x2+1) as u.

u=x2+1 (1)

Differentiate both sides of the Equation (1).

du=2xdx (2)

Rearrange Equation (2) to find the value of x2 as shown below.

x2=u1 (3)

Substitute u for (x2+1), (u1) for x2, and 12du for xdx in the function as shown below.

x3x2+1dx=(u1)u(12du)=12(u11/21u)du=12(u1u)du (4)

The expression to find the indefinite integral value using Equation (4) as shown below.

x3x2+1dx=12(u1u)du=12(u12+1(12+1)u12+1(12+1))=12(u3/232u1/212)=u3/23u1/2 (5)

Substitute (x2+1) for u in Equation (5) as shown below

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