# Homework Help: Improper integral x^(2) * e^(-x^2)

1. Aug 26, 2010

### TsAmE

1. The problem statement, all variables and given/known data

Show that $$\int_{0}^{\infty}x^{2}e^{-x^{2}}dx = \frac{1}{2}\int_{0}^{\infty}e^{-x^{2}}dx.$$

2. Relevant equations

None.

3. The attempt at a solution

I used substitution:

$$t = x^{2}$$

$$dx = \frac{dt}{2x}$$

$$\frac{1}{2}\int_{0}^{\infty}\sqrt{t}e^{-t}dx$$

Then tried using integration by parts but then I didnt get an answer and got stuck.
1. The problem statement, all variables and given/known data

2. Relevant equations

3. The attempt at a solution

2. Aug 26, 2010

### lanedance

have you tried straight integartion by parts?

3. Aug 26, 2010

### gomunkul51

Try:

x^2*e^(-x^2) = [-0.5*x]*[-2*x*e^(-x^2)dx]

[-0.5*x] = u
[-2*x*e^(-x^2)dx] = dv

Tell us if you got it !

Good Luck :)

4. Aug 26, 2010

### TsAmE

I tried integration by parts but it didnt work out. I only learn how to integrate e^(-x^2) next year.

5. Aug 26, 2010

### HallsofIvy

How did you try? Did you try letting $u= x$ and $dv= xe^{-x^2}dx$.

6. Aug 26, 2010

### gomunkul51

Try integration by parts with t=x, not t=x^2 !

I remind you:

integral(u*dv) = u*v - integral(v*du)

P.S.: If you learn in the next year how to get the anti-derivative of e^(-x^2), please tell us ;)

Last edited: Aug 26, 2010