text: 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
  
  ### Still sussing out bartleby?
  
  Check out a sample textbook solution.
  
  See a sample solution
  
  #### The Solution to Your Study Problems
  
  Bartleby provides explanations to thousands of textbook problems written by our experts, many with advanced degrees!
  
  Get Started