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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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    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.2012-09-05

1 Answers 1

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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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    No more help needed, I got it. Thank you!2012-09-05