Prove that $\int_{-\infty}^{\infty}\frac{\sin\omega \cos{2\omega}}{\omega}d\omega=0$

Question

While studying Fourier Integral topic, I encountered this problem. I have to prove that

$$\int_{-\infty}^{\infty}{\frac{\sin\omega \cos{2\omega}}{\omega}} d\omega =0$$

I've no idea how to do this. Could someone guide me, please?

Answer 1

HINT $$\sin(x)\cos(y) = \frac{\sin(x+y) + \sin(x-y)}{2}$$