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

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?

• Hint : convert the product to sum. – Zaid Alyafeai Mar 18 '17 at 9:38

HINT $$\sin(x)\cos(y) = \frac{\sin(x+y) + \sin(x-y)}{2}$$
• so, $\int_{-\infty}^{\infty}{\frac{\sin\omega \cos{2\omega}}{\omega}}=\frac{1}{2}\left(\int_{-\infty}^{\infty}{\sin(3\omega)} + \int_{-\infty}^{\infty}{\sin(-\omega)}\right)$, but these integrals does not converge. – 01000110 Mar 18 '17 at 10:06
• You was left $\frac{1}{w}$ – kotomord Mar 18 '17 at 10:12