   Chapter 6, Problem 17TYS ### 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
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# In Exercises 14–17, determine whether the improper integral diverges or converges. Evaluate the integral if it converges. ∫ 0 ∞ 2 x 3 e − x 4 d x

To determine

The value of the improper integral 02x3ex4dx if possible.

Explanation

Consider the provided integral:

02x3ex4dx

From definition of improper integral.

af(x)dx=limbabf(x)dx

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

eax=eaxa+C

Use the property of improper integral and simplify as:

02x3ex4dx=limb0b2x3ex4dx

Integrate the integrand by substation method as:

Assume x4=u

Differentiate as:

4x3dx=du

Now, substitute the values and integrate by using the inverse function formula as:

2xex2dx=12eudu=12eu+C

Again, substitute the value of u

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