I am unsure if this belongs in calculus, differential equation, advance applied math, etc.

So I would like to figure out the trick to

$\displaystyle

g(x) = \frac{1}{2\pi}\int_{-\infty}^{+\infty}\! \sin(c \omega t) e^{-i\omega x} \, d\omega$.

Yes, I know this is an Inverse Fourier Transform, and I know there are tables. What I am looking for is a good explanation of how

$\displaystyle

g(x) = \frac{i}{2}\left[\delta(x+ct) - \delta(x-ct)\right]$.

Also, I figure this might help me understand the Dirac-delta function a bit better, because I am unsure why

$\displaystyle f(x) \ast g(x) = f(x + ct) - f(x - ct)$.

Thank you in advance.