Integrating x^2 * e^-x^2 from -Infinity to 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

Homework Equations

The Attempt at a Solution

I really don't 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

 

[phyzguy]

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.

 

[phyzguy]

Try writing x e^-x2 as x / e^x2, then expand the e^x2 in the denominator as a power series and watch what happens as x goes to infinity. The e^x2 term grows much faster than any power of x.