# Prove that: $\lim_{n\to\infty}\int_0^{\pi/2} \frac{\cos(nx)}{n^2+x^2}dx=0$

How do I prove that the following is true:

$$\lim_{n\to\infty}\int_0^{\pi/2} \frac{\cos(nx)}{n^2+x^2}dx=0$$

Hint: Note that $-1\leq \cos nx\leq 1$ and hence we can get suitable bounds for the integral and now use Squeeze Theorem. Answer should be $0$.