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 $\sqrt{2\pi}$.

$$ \Large{\color{#66f}{\int_{0}^{\infty} x^{-\frac{1}{2}} e^{-\frac{x}{2}} dx}}$$
$$\bbox[8pt,border:2pt solid #66f]{\begin{matrix}{\color{crimson}{\text{Substitute:}}\\ u=x/2 \implies du= \frac{1}{2} dx }\end{matrix}}$$
$$\color{crimson}{-------------------}$$
$$\text{The Integral Becomes:}$$
$$2 \int_0^{\infty} {x}^{-\frac{1}{2}} e^{-u} du$$
$$2\int_0^{\infty} (2u)^{-\frac{1}{2}} e^{-u} du = \sqrt{2} {\int_0^{\infty} u^{-\frac{1}{2}} e^{-u} du}=\sqrt{2} \cdot \color{#ff6600}{\Gamma\left(\frac{1}{2}\right)}$$
Where $\Gamma(n)$ is the Gamma Function
$$\bbox[8pt,border:2pt solid crimson]{\color{#ff6600}{\Gamma\left(\frac{1}{2}\right)}=\sqrt{\pi}}$$
$$(\spadesuit)$$
$$ \Large{\int_{0}^{\infty} x^{-\frac{1}{2}} e^{-\frac{x}{2}} dx=\sqrt{2\pi}}$$
