   Chapter 13, Problem 34RE ### Mathematical Applications for the ...

11th Edition
Ronald J. Harshbarger + 1 other
ISBN: 9781305108042

#### Solutions

Chapter
Section ### Mathematical Applications for the ...

11th Edition
Ronald J. Harshbarger + 1 other
ISBN: 9781305108042
Textbook Problem

# Evaluate the improper integrals in Problems 31-34. ∫ − ∞ 0 x ( x 2 + 1 ) 2 d x

To determine

To calculate: The value of the improper integral 0x(x2+1)2dx if it converges.

Explanation

Given Information:

The provided integral is,

0x(x2+1)2dx

Formula used:

According to the power rule of integrals,

xndx=xn+1n+1+C

If the limit defining the improper integral is a unique finite number, then integral converges else it diverges,

bf(x)dx=limaabf(x)dx

Calculation:

Consider the provided integral,

0x(x2+1)2dx

Now, use the formula,

bf(x)dx=limaabf(x)dx

To rewrite the integral by multiplying and dividing it by 2 as,

0x(x2+1)2dx=12limaa02x(x2+1)2dx

Now, let x2+1=t, then on obtaining differentials,

2xdx=dt

Thus, the integral becomes,

0x(x2+1)2dx=12limaa02x(x2+1)2dx=12limaa0dtt2

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