Recently I had to use the fact that the Dirichlet integral evaluates as
$\int_{0}^{\infty} \frac{\sin{x}}{x} \ dx = \frac{\pi}{2}$ a couple of times.
There already is a question that specifically ask for methods to show this result $\textbf{not}$ using complex integration. In this question I am interested in seeing the derivation via contour integration. ( I am aware of the wikipedia entry, but am looking for more detail )