# Closed form solution of a definite integral?

I've never heard of a closed form solution, and I had this assigned to me and it is supposed to be 'review'. Could somebody at least get me pointed in the right direction here?

'Provide a closed form solution for the following definite integrals'

$$\int_0^\infty xe^{-x^2}\ dx$$

$$\int_0^\infty x^2e^{-x}\ dx$$

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Post a link to the pictures in the comments. We'll be able to edit it into your post. – user2468 Sep 5 '12 at 14:59
i.stack.imgur.com/RZeQY.png – Hoser Sep 5 '12 at 14:59
Why my edit is not accepted? – Zeta.Investigator Sep 5 '12 at 15:03
Sorry, just accepted it. – Hoser Sep 5 '12 at 15:07
My guess is the professor asked for a "closed form" to prevent students from turning in the first step of integration by parts ($uv - \int v du$) and proclaiming the problem to be solved. Such a solution is still "open" in the sense that there is yet another integral to evaluate. – Austin Mohr Sep 5 '12 at 15:39

For the first question, make the substitution $u=x^2$. After a while you should get $1/2$.
For the second, integrate by parts twice. For the first integration by parts, let $u=x^2$ and $dv=e^{-x}\,dx$.
It's been a while since doing IBP, for the second one I ended up getting $-e^{-x}\(x^2 - 2x) - 2$, does that seem correct? – Hoser Sep 5 '12 at 15:26
There is a little sign error, it should be $-e^{-x}(x^2+2x+2)$. Minus signs are a pain: if one does a couple of integrations by parts, probability of a sign error is high. Note that you can always check an integral by differentiating. – André Nicolas Sep 5 '12 at 15:32