$k \in \mathbb N$. I know, that this is an odd function, if $k$ is an even number. So I have to calculate the integral from $-\infty$ to $+\infty$ and that would be $0$ for $k$ as an even number. But how do I calculate the integral for an odd number of $k$. Wolfram Alpha says something like:

$$\int_0^\infty \exp\left(-\frac{1}{2}x^2\right)x^k \, \mathrm{d}x = 2^{\frac{k-1}{2}} \Gamma\left(\frac{k+1}{2}\right)$$

But how do I get there? Or is there a easier method?

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    $\begingroup$ Sub $x=\sqrt{2 u}$ and use the definition of the Gamma function. $\endgroup$ – Ron Gordon Nov 11 '15 at 21:53
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    $\begingroup$ Compute $\int e^{-ax^2/2}=\sqrt{\pi/a}$ by squaring and converting to polar coordinates. Then take the derivative of the answer with respect to $a$ $k/2$ times and set $a=1$. $\endgroup$ – ziggurism Nov 11 '15 at 22:06

Hint: Let $t=\dfrac{x^2}2,~$ and then employ the well-known integral expression for the $\Gamma$ function.


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