   Chapter 13.7, Problem 17E ### 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

# In Problems 1-20, evaluate the improper integrals that converge. 17.   ∫ − ∞ ∞ 2 x ( x 2 + 1 ) 2   d x

To determine

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

Explanation

Given Information:

The provided integral is,

2x(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,

f(x)dx=limaacf(x)dx+limbcbf(x)dx

Calculation:

Consider the provided integral,

2x(x2+1)2dx

Now, use the formula,

f(x)dx=limaacf(x)dx+limbcbf(x)dx

To rewrite the integral as,

2x(x2+1)2dx=limaa02x(x2+1)2dx+limb0b2x(x2+1)2dx

Let x2+1=t, then differentiate both sides with respect to x,

2xdx=dt

Thus, the integral becomes,

2x(x2+1)2dx=limaa02x(x2+1)2dx+limb0b2x(x2+1)2dx=limaa0dtt2+limb0bdtt2

Now, use the power rule of integrals to evaluate the integral,

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