Prove these integrals are equal

**paupsers** · February 7th 2010, 01:16 PM

Prove that

$\int_{0}^\infty x^2 e^{{-x}^{2}}=\frac{1}{2} \int_{0}^\infty e^{{-x}^{2}}$

and both integrals converge.

Any help on this? I tried doing integration by parts for the second one but couldn't reach an answer.

**Jhevon** · February 7th 2010, 01:38 PM

Originally Posted by paupsers

Prove that

$\int_{0}^\infty x^2 e^{{-x}^{2}}=\frac{1}{2} \int_{0}^\infty e^{{-x}^{2}}$

and both integrals converge.

Any help on this? I tried doing integration by parts for the second one but couldn't reach an answer.

see post #2 here

**paupsers** · February 7th 2010, 02:23 PM

That thread was really helpful!

But now I'm trying to solve

$\int_0^\infty x^2e^{-x^2}dx$ by using the same method, and it's not working out so nicely. I've set it up the same way and converted it into polar, and now I'm at

$\int_0^\infty\int_0^{2\pi}r^5cos^2\theta sin^2\theta e^{-r^2}d\theta dr$ and I have a feeling that's not simple to evaluate... :-(

Is there a simpler way to do this?

**paupsers** · February 7th 2010, 02:37 PM

Actually, it can be done using integration by parts. Thanks for the link, still!

Thread: Prove these integrals are equal

LinkBack

Thread Tools

Display

Prove these integrals are equal

Similar Math Help Forum Discussions

Prove me that pi is no equal 2

Show two integrals are equal

Prove two double integrals are equal

Proof involving integrals equal to 0.

prove H_T less than or equal to S_n

Search Tags