   Chapter 13.7, Problem 14E ### 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. 14.   ∫ − ∞ 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

Multiply and divide by 2 to rewrite the integral as,

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

Now, let x2+1=t, and then differentiate both the sides,

2xdx=dt

Thus, the integral becomes,

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

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