   Chapter 7.1, Problem 45E

Chapter
Section
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 3 1 + x 2   d x

To determine

To evaluate: The given integral and verify by plotting both the function and its antiderivative

Explanation

The technique of integration by parts comes in handy when the integrand involves product of two functions. It can be thought of as a rule corresponding to the product rule in differentiation.

Formula used:

The formula for integration by parts in terms of u and v is given by

udv=uvvdu

Given:

The integral, x31+x2dx.

Calculation:

Rewrite the integral using the substitution 1+x2=t, then 2xdx=dt. The integration in terms of t will be

x31+x2dx=12x21+x22xdx=12(t1)tdt=12(t32t)dt

Integrate further to get:

x31+x2dx=12(t32t)dt=12(25t5223t32)+C=t3

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