Solving Integration Problem: Finding Int. of x^2e^(-x^2)

[qspeechc]

I feel really dumb for asking this, because I know it's something simple I'm just not seeing. Ok, given that

$$\int _{-\infty}^{\infty} e^{-x^2}dx = \sqrt{\pi }$$

how to I find

$$\int _{-\infty}^{\infty} x^2e^{-x^2}dx = ?$$

I have tried the substitution u=x^2, and integration by parts, but nothing is working. Any help? Thanks

 

[Dick]

The easy way to do this problem is to generalize your first integral. Can you show the integral of exp(-ax^2) is sqrt(pi/a) for a>0? (Use a substitution x=sqrt(a)*u). Now differentiate that with respect to a. Finally put a=1 again.

 

[rootX]

This might help:

d/dx (x*[e^-x^2]) = ...
Solve it and then integrate!

 

[statdad]

Integration by parts should work. You have

$$\int_{-\infty}^\infty x^2 e^{-x^2} \, dx = \sqrt \pi$$

Set

$$u = x, \quad dv =x e^{-x^2} dx$$

Then

$$\int u \, dv = uv - \int v \, du$$

should, with careful attention to the [tex]uv $$term at the infinities, work fine.$$[/tex]

 

[qspeechc]

Thanks everyone.