The function you have to integrate is even, so the integral on is two times the integral on .
Another way: denote , then . Using the substitution we get (Dirichlet's integral) . So, . For we get .
@fareastmovement: Show your complete work by the method of residues (of course if you want) and we can check it.
Edited: Method not valid, see the posts below.
Fubini's theorem is not satsified. So to justify changing the order of integration, let's write the double integral more formally as a limit.
Since the Sine Integral is bounded above by 2, we can use the Dominated Convergence Theorem to justify bringing the limit inside of the integral.