Gaussian Integral (half factorial)
Integration done by converting the integration wrt cartesian coordinates → polar coordinates.
$$ \begin{align} I=\int \limits_0^\infty e^{-x^2}dx = (\frac{1}{2})!=\frac{\sqrt{\pi}}{2} \\ Proof: \\ I^2 = \int \limits_0^\infty e^{-x^2}dx \cdot \int \limits_0^\infty e^{-y^2}dy\\ =\iint \limits_0^\infty e^{-(x^2+y^2)}dxdy\\ =\int \limits_{r=0}^{r=\infty} \int \limits_{\theta=0}^{\theta=\frac{\pi}{2}} e^{-r^2}rd\theta dr\\ =\frac{\pi}{2}\cdot-\frac{1}{2} \int \limits_{r=0}^{r=\infty}e^{(-r^2)}(-2rdr)\\ =\frac{\pi}{2}\cdot-\frac{1}{2} [e^{-r^2}]_0^\infty=\frac{\pi}{4}\\ \\ I=\frac{\sqrt{\pi}}{2} \end{align} $$