How to prove $\int^{\infty}_{0}\frac{\sin x}{x}\mathrm{d}x=\frac{\pi}{2}$?

Question

Possible Duplicate:
Solving the integral $\int_{0}^{\infty} \frac{\sin{x}}{x} \ dx = \frac{\pi}{2}$?

How to prove $$\int^{\infty}_{0}\frac{\sin x}{x}\mathrm{d}x=\frac{\pi}{2}$$ Do you have a simple method ?

Answer 1

This is known as the Dirichlet integral. A couple of different derivations appear on the Wikipedia page:

http://en.wikipedia.org/wiki/Dirichlet_integral#Evaluation