Chapter 14, Problem 9PS

### Calculus (MindTap Course List)

11th Edition
Ron Larson + 1 other
ISBN: 9781337275347

Chapter
Section

### Calculus (MindTap Course List)

11th Edition
Ron Larson + 1 other
ISBN: 9781337275347
Textbook Problem

# Evaluating an IntegralIn Exercises 9 and 10, evaluate the integral. (Hint: See Exercise 63 in Section 14.3.) ∫ 0 ∞ x 2 e − x 2 d x

To determine

To calculate: The value of integral 0x2ex2dx.

Explanation

Given: The integral âˆ«0âˆžx2eâˆ’x2dx.

Formula used: The gauss error function is aspecial function defined as

erf(x)=2Ï€âˆ«eâˆ’x2dx

Where, erf(âˆž)=1 and erf(0)=0.

Integration by parts:

âˆ«uâ€‰dv=uâ‹…vâˆ’âˆ«uâ€²v

Calculation: The integral is âˆ«0âˆžx2eâˆ’x2dx that can be rewritten as âˆ«0âˆžxâ‹…(xâ€‰eâˆ’x2)â€‰dx.

Now, let u=x and dv=xâ€‰eâˆ’x2.

So, uâ€²=1 and v=âˆ’eâˆ’x22.

Now, by use integration by parts the integral is:

âˆ«0âˆžx2eâˆ’x2dx=[xâ‹…(âˆ’eâˆ’x22)âˆ’âˆ«1

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