Which methods can be used to evaluate the following integral? How can I evaluate the following integral
$$ \int_{0}^{\infty} x^{-1/2} \exp({-x/2})\ dx $$
I know the answer is $\sqrt{2\pi}$.

The standard elementary proof is set $u = \sqrt{x}$, convert it to the square root of a 2-d integral and then evaluate the 2-d integral in polar coordinates:
$$
\int_0^\infty x^{-1/2} e^{-x/2} dx
= 2\int_0^\infty e^{-u^2/2} du = \int_{-\infty}^\infty e^{-u^2/2} du\\
= \sqrt{ \int_{-\infty}^{\infty} \int_{-\infty}^{\infty}  e^{-(u^2+v^2)/2} dudv}
= \sqrt{ 2\pi \int_0^{\infty} e^{-r^2/2} rdr} = \sqrt{2\pi}
$$
