Integrating functions I've never dealth with

Question

If $a>0$ show that

$$\lim_{n \to \infty} \int_a^\pi \frac{\sin(nx)}{nx} dx = 0.$$

I've never dealt with non-elementary integral functions before and I'm not sure why this would show up on a study guide the week of finals. I suppose this is Riemann integrable, but I have no idea how to show that.

Answer 1

Assume $0<a<\pi$. One may write, as $n \to \infty$, $$ \left|\int_a^\pi\frac{\sin(n x)}{n x} dx\right|\leq \int_a^\pi\frac{\left|\sin(n x) \right|}{n x}dx\leq \frac{\pi-a}{n a}\to 0. $$

Answer 2

Hint: $\sin(nx)$ is bounded above by $1$ and below by $-1$. So, the integrand is bounded above by $\frac{1}{nx}$ and below by $\frac{-1}{nx}$. Now, if you integrate these on the interval $a$ to $\pi$, you'll get an upper and lower bound on the integral. When you let $n \to \infty$, the bounds will coincide to $0$.