Gaussian integral

The Gaussian integral is:
\begin{align}
\int_{-\infty}^{\infty}dx e^{-x^2} \,=\, \sqrt{\pi}
\end{align}
A standard way to derive this formula is
\begin{align}
\bigg(\int_{-\infty}^{\infty}dx e^{-x^2}\bigg)^2
\,&=\, \int_{-\infty}^{\infty}dx \int_{-\infty}^{\infty}dy\, e^{-(x^2+y^2)}
\\
\,&=\, \int_0^{2\pi} d\phi \int_{0}^{\infty}dr\,r e^{-r^2}
\\
\,&=\, \int_0^{2\pi} d\phi \int_{0}^{\infty}{dr^2 \over 2}\, e^{-r^2}
\\
\,&=\, \pi
\end{align}
Performing the change of variable \(x=\sqrt{t}\), this turns into the Euler integral
\begin{align}
\int_{-\infty}^{\infty}dx\, e^{-x^2} \,=\, 2 \int_0^\infty dx\, e^{-x^2}
\,=\, 2\int_0^\infty dt\, \frac{1}{2}\ e^{-t} \ t^{-\frac{1}{2}}
\,=\, \Gamma\left(\tfrac{1}{2}\right) = \sqrt{\pi}
\end{align}
where \(\Gamma\) is the gamma function. From this we can see that the factorial of a half-integer is a rational multiple of \(\sqrt{\pi}\). More generally, let us consider
\begin{align}
\int_0^{\infty} dx\, e^{-\alpha x^\beta}
\end{align}
Change the varible \(t=\alpha x^\beta\),
\begin{align}
dx \,=\, {1 \over \alpha^{1\over \beta} \beta}\, t^{{1\over\beta} – 1}\, dt
\end{align}
and thus we have
\begin{align}
\int_0^{\infty} dx\, e^{-\alpha x^\beta}
\,=\, {1 \over \alpha^{1\over \beta} \beta}\,\int_0^{\infty} dt\, t^{{1\over\beta} – 1}\,e^{-t} dt
\,=\, {\Gamma\left(\tfrac{1}{\beta}\right) \over \alpha^{1\over \beta} \beta}
\end{align}

Another type of integrals is
\begin{align}
2\int_0^{\infty} dx\,x^{2n} e^{-x^2}
\,=\, \int_0^{\infty} dt\,t^{n-{1\over 2}} e^{-t}
\,=\, \Gamma(n+\tfrac{1}{2})
\end{align}

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