Prove that $\lim_{n \rightarrow \infty} \int_{0}^{2 \pi} \frac {\sin (nx)}{x^2 + n^2} dx = 0$.

Question

Prove that $\lim_{n \rightarrow \infty} \int_{0}^{2 \pi} \frac {\sin (nx)}{x^2 + n^2} dx=0.$

I think I will use Riemann integrability, but how I do not know, could anyone help me in this?

Answer 1

$|\frac {\sin\, x} {x^{2}+n^{2}}| \leq \frac 1 {n^{2}}$.

Answer 2

HINT

Recall that for $f$ continuous on $(a,b)$

$$\left|{\int_a^b f \left({t}\right) \ \mathrm dt}\right| \le \int_a^b \left|{f \left({t}\right)}\right| \ \mathrm dt$$

therefore we have that

$$0\le \left|\int_{0}^{2 \pi} \frac {\sin (nx)}{x^2 + n^2} dx\right|\le \frac1{n^2}\int_{0}^{2 \pi} \frac {1}{(x/n)^2 + 1} dx =\frac{1}{n}\arctan\left(\frac{2\pi}n\right)$$