Use integration by parts to evaluate the integral . Note that evaluation may require integration by parts more than once.

Question

Use integration by parts to evaluate the integral  . Note that evaluation may require integration by parts more than once. ​

Quizplus · Accepted Answer

Using integration by parts with u = x^2 and dv = e^(-x^2)dx, we get:
 
v = (-1/2)√π erf(x)
du/dx = 2x
 
Then, applying integration by parts again with u = 2x and dv = e^(-x^2)dx, we get:
 
v = (-1/2)√π erf(x)
du/dx = 2
 
Putting it all together and applying the limits of integration gives:
 
∫[0,∞] x^2e^(-x^2)dx = [-x e^(-x^2)/2 - √π/4 erf(x)]|[0,∞]
 
= [-∞ - 0 + (√π/4)(2)] - [0 - 0 + (√π/4)(0)]
 
= (√π/2)

Use Integration by Parts to Evaluate the Integral