Chapter 13.7, Problem 12E

### Mathematical Applications for the ...

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

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. 12.   ∫ − ∞ − 2 x ( x 2 − 1 ) 3   d x

To determine

To calculate: The value of the improper integral 2x(x21)3dx if it converges.

Explanation

Given Information:

The provided integral is,

2x(x21)3dx

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,

2x(x21)3dx

Now, use the formula,

bf(x)dx=limaabf(x)dx

Multiply and divide by 2 and rewrite the integral as,

2x(x21)3dx=12limaa22x(x21)3dx

Now, let x21=t, then differentiate both the sides,

2xdx=dt

Thus, the integral becomes,

2dtt3dx

Now evaluate the integral,

2x(x21)3dx=12limaa22x(x21)3dx=12limaa2dtt3

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