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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 – 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
up vote 4 down vote accepted

Hint: It is a fancy way of asking you to compute the two integrals.

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 $dv=e^{-x}\,dx$.

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So are you saying I should be using integration by parts for both, or just the second? I'm not really sure what a closed form solution is aside from what I've been able to read up on it by searching online. – Hoser Sep 5 '12 at 15:09
Nevermind, I see where you said it is a 'fancy way of asking to compute the integrals'. Thank you! This was very helpful. – Hoser Sep 5 '12 at 15:10
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
Alright, thanks. Disregard what I had typed here before, I had gotten confused. – Hoser Sep 5 '12 at 15:34

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