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Mathematical Applications for the ...

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

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BuyFindarrow_forward

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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Sect-13.1 P-9ESect-13.1 P-10ESect-13.1 P-11ESect-13.1 P-12ESect-13.1 P-13ESect-13.1 P-14ESect-13.1 P-15ESect-13.1 P-16ESect-13.1 P-17ESect-13.1 P-18ESect-13.1 P-19ESect-13.1 P-20ESect-13.1 P-21ESect-13.1 P-22ESect-13.1 P-23ESect-13.1 P-24ESect-13.1 P-25ESect-13.1 P-26ESect-13.1 P-27ESect-13.1 P-28ESect-13.1 P-29ESect-13.1 P-30ESect-13.1 P-31ESect-13.1 P-32ESect-13.1 P-33ESect-13.1 P-34ESect-13.1 P-35ESect-13.1 P-36ESect-13.1 P-37ESect-13.1 P-38ESect-13.1 P-39ESect-13.1 P-40ESect-13.2 P-1CPSect-13.2 P-2CPSect-13.2 P-1ESect-13.2 P-2ESect-13.2 P-3ESect-13.2 P-4ESect-13.2 P-5ESect-13.2 P-6ESect-13.2 P-9ESect-13.2 P-10ESect-13.2 P-11ESect-13.2 P-12ESect-13.2 P-7ESect-13.2 P-8ESect-13.2 P-23ESect-13.2 P-24ESect-13.2 P-13ESect-13.2 P-14ESect-13.2 P-15ESect-13.2 P-16ESect-13.2 P-17ESect-13.2 P-18ESect-13.2 P-19ESect-13.2 P-20ESect-13.2 P-21ESect-13.2 P-22ESect-13.2 P-25ESect-13.2 P-26ESect-13.2 P-27ESect-13.2 P-28ESect-13.2 P-29ESect-13.2 P-30ESect-13.2 P-31ESect-13.2 P-32ESect-13.2 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