Solution
= Solution
In the <Gamma function> integral at $p=1/2$, use $x=t^2$, $dx=2t\,dt$, with $t>0$. Then $x^{-1/2}dx=2\,dt$, so
$$
\Gamma(1/2)=\int_0^\infty x^{-1/2}e^{-x}dx=2\int_0^\infty e^{-t^2}dt.
$$
The substitution can first be made on finite positive intervals and then extended by monotone limits, justifying the endpoint passage.