Elegant integration of sinx/x from 0 to infinity

I learned two things called Contour integral and Jordan's lemma today.

It helps solve this integral elegantly.

To evaluate this you can use e^iz/z and integrate it over 2 semi circles and 2 line segments and residue theorem to convert this into evaluation of this integral. It is really elegant.
It uses Jordan's lemma along the way to make one of the semi circle integral 0 as its radius R goes to infinity.