   Chapter 13.7, Problem 6E ### 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. 6.   ∫ 0 ∞ x 2 e − x 3   d x

To determine

To calculate: The value of the improper integral 0x2ex3dx if it converges.

Explanation

Given Information:

The provided integral is,

0x2ex3dx

Formula used:

According to the exponential rule of integrals,

exdx=ex+C

If the limit defining the improper integral is a unique finite number, then integral converges else it diverges,

af(x)dx=limbabf(x)dx

Calculation:

Consider the provided integral,

0x2ex3dx

Now, use the formula,

af(x)dx=limbabf(x)dx

Also, multiply and divide the integral by 3 and rewrite as,

0x2ex3dx=13limb0b3x2ex3dx

Now, let x3=t,

Differentiate both the sides,

3x2dx=dt

Therefore, the integral becomes,

0x2e

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