Prove that ∫₀^∞ x²e^(-ax²) dx = (1/4)√(π/a³)

[henryc09]

Homework Statement

Given that:

The integral between infinity and -infinity of

e^{-ax^2} dx = $$\sqrt{\pi/a}$$

show that

The integral between 0 and infinity of

x².e^{-ax^2} dx = 1/4$$\sqrt{\pi/a^3}$$

Homework Equations

The Attempt at a Solution

don't really know where to start

 

[lurflurf]

integrate by parts