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[itex]psi=ae^{-\frac{m\omega}{2\hbar}x^2}[/itex]

I have been trying to integrate [itex]\psi ^2[/itex] to find the constant a so that the wave function is normalised, and I know the trick with converting to polar coordinates to integrate [itex]e^{-x^2}[/itex], but I cannot figure out how to integrate the more complicated version above. I know that the constant should have the value [itex]\left(\frac{m\omega}{\pi \hbar}\right)^{\frac{1}{4}}[/itex] if the wavefunction is to be normalised, but I can't figure out how to do this?

Thank you in advance!