Integrating from - to + infinity

[terp.asessed]

Homework Statement

I am having trouble integrating ∫ (x = -∞ to +∞) x³e^-αx2 dx part--is this 0 or 1/α²? And, could someone explain? I am pretty sure that, when

∫ (x =0 to +∞) x³e^-αx2 dx = 1/α²

However, with x = -∞ to +∞, and the function of the equation being odd, I am lost.

Homework Equations

above

The Attempt at a Solution

above

 

[vela]

It's zero because, like you noted, the integrand is odd.