   Chapter 13.7, Problem 18E ### 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. 18.   ∫ − ∞ ∞ 9 x 5 ( 3 x 6 + 7 ) 2   d x

To determine

To calculate: The value of the improper integral 9x5(3x6+7)2dx if it converges.

Explanation

Given Information:

The provided integral is,

9x5(3x6+7)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,

9x5(3x6+7)2dx

Now, use the formula,

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

And multiply and divide by 2 to rewrite the integral as,

9x5(3x6+7)2dx=12limaa018x5(3x6+7)2dx+limb0b18x5(3x6+7)2dx

Let 3x6+7=t, then differentiate both the sides with respect to x,

18x5dx=dt

Thus, the integral becomes,

9x5(3x6+7)2dx=12limaa018x5(3x6+7)2dx+limb0b18x5(3x6+7)2dx=12limaa0dtt2+limbb0dtt2

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

9x5(3x6+7)2dx=12[limaa018x

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