# Improper Integral Infinity

phrygian

## Homework Statement

Using the fact that the integral from -Infinity to Infinity of e^-x^2 is equal to Sqrt(Pi), find the integral from -Infinity to Infinity of x^2 * e^-x^2

## The Attempt at a Solution

I really dont know how to find this using the fact that the first integral is equal to Sqrt(Pi), where do you start on this one?

Thanks for the help

Phrygian,

Try integrating by parts with u = x, and dv = x e-x2 dx.

phrygian
Thanks a lot! But now how do I evaluate -x/2(e^-x^2) from -Infinity to Infinity?

Count Iblis
The derivative of the argument of the exponential function is, up to a constant factor, in front of the exponential function.

phrygian
After doing the integration by parts I ended up with -x/2(e^-x^2) to be evaluated from -infinity to infinity + integral of 1/2 e^-x^2 dx from negative infinity to infinity. I know that the second integral is equal to Sqrt(Pi)/2 but I can't figure out how to evaluate the first part at the limits.