How would I calculate gamma function of $\frac12$ by hand?

Question

There are almost no tutorials how to compute the gamma function by hand. If I try to do it myself, I fall into all sorts of pitholes, for example finding a limit that is indeterminate no matter how many times I differentiate the both sides of the fraction, or integrating by parts in an infinite loop, etc.

There must be some tricks to calculate the gamma function by hand.

So how I would for example compute the gamma function of $\frac12$?

Answer 1

$$\Gamma(1/2)=\int_0^\infty x^{-1/2}\exp(-x)dx.$$

Let $u=\sqrt{x}$. We get:

$$\int_0^\infty \frac{1}{u}e^{-u^2}2udu=2\int_0^\infty e^{-u^2}du.$$

The last integral is quite famous and easy to compute. Can you finish it from here?