Find the exact value of Integral from 0 to infinity of x * f(x) dx?

Question

Find the exact (not calculator) value of Integral from 0 to infinity of x * f(x) dx, given the fact that Integral from 0 to infinity of e^(-x^2) dx = sqrt(pi)/2

f(x) = xe^(-x^2)

Answer 1

https://www.wolframalpha.com/input/?i=+%7Bx%3A+0+t...

Answer 2

∫ x * f(x) = ∫ x * xe^(-x²) =  ∫ x²e^(-x²) 

Now hint: Do this by parts. What do you get?