   Chapter 6.4, Problem 20E ### Calculus: An Applied Approach (Min...

10th Edition
Ron Larson
ISBN: 9781305860919

#### Solutions

Chapter
Section ### Calculus: An Applied Approach (Min...

10th Edition
Ron Larson
ISBN: 9781305860919
Textbook Problem
1 views

# Evaluating an Improper Integral In Exercises 7–20, determine whether the improper integral diverges or converges. See Example 1, 2, and 3. ∫ − ∞ ∞ x 2 e − x 3 d x

To determine

To calculate: Whether the improper integral x2ex3dx diverges or converges and evaluate if it converges.

Explanation

Given Information:

The integral is provided as:

x2ex3dx

Formula used:

From definition of improper integral.

f(x)dx=limaa0f(x)dx+limb0bf(x)dx

Also, the integral for the integration of a inverse function is as follows:

eax=eaxa+C

Calculation:

Consider the provided integral:

x2ex3dx

Use the property of improper integral f(x)dx=limaa0f(x)dx+limb0bf(x)dx and simplify as:

x2ex3dx=limaa0x2ex3dx+limb0bx2ex3dx

Integrate the integrand by substation method as:

Assume x3=u

Differentiate as:

3x2dx=du

Now, substitute the values and integrate by using the inverse function formula eax=eaxa+C

as:

x2ex3dx=13eudu=13eu+C

Again, substitute the value of u

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