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I arrive at the partial solution of $((x*e^{-x^2})/2)- (\pi^{1/2}/2)$ using double integration by parts. How do I resolve the first part of the solution with e. Is the solution even right?

amWhy
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user218197
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    I fixed the title of your question with LaTex. Please apply the same for the body of the question (it's not quite clear what you mean there with all the parenthesis). – barak manos Feb 21 '15 at 18:57
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    basically a duplicate of http://math.stackexchange.com/q/1149907/173147 (see also the answers on the general cases) – glS Feb 21 '15 at 18:59
  • I voted to close as a duplicate because the same process(es) needed to evaluate this integral is/are demonstrated fully in the post of which this is essentially a duplicate. – amWhy Feb 21 '15 at 19:30

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$$\int_{-\infty}^{\infty} x^2 e^{-x^2} \, dx = 2 \int_0^{\infty}x^2 e^{-x^2} \, dx = 2J. $$

$\begin{align} J &= \int_0^{\infty}x^2 e^{-x^2} \, dx\\ &= \int_0^{\infty} x d(-\frac12e^{-x^2}) = x (-\frac12e^{-x^2})\Big|_0^\infty+\frac12\int_0^{\infty}e^{-x^2}\, dx\\ &=\frac12\int_0^{\infty}e^{-x^2}\, dx\\ &= \frac{\sqrt \pi}4 \end{align}$

that gives us $$\int_{-\infty}^{\infty} x^2 e^{-x^2} \, dx = \frac{\sqrt \pi}2$$

abel
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