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I'm wondering how to solve the following definite integral,

[itex]\int^\infty_{-\infty}{x^4e^{-x^2}dx}[/itex].

I know the answer is ##\frac{3 \sqrt{\pi}}{4}##, but I'm not positive how to get there.

I understand how to evaluate the definite "Gaussian" integral $$\int^\infty_{-\infty}{e^{-x^2}}=\sqrt \pi$$ using a switch to polar coordinates and a u sub, but not sure if/how that applies here.

Extending the question, I'm also wondering about an integral like $$\int^\infty_{-\infty}{e^{ikx}e^{-x^2}}$$.

Thanks!

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# Integrals/Non-Elementary Antiderivatives

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