How to evaluate $\lim_{n\rightarrow \infty }\left ( \int_{0}^{2\pi}{\frac{\cos(nx)}{x^{2}+n^{2}}}dx \right ) where\ \forall \ \ n\in \mathbb{N} $?

Any Idea how to integrate the Integral in the brackets and than applying limits:

$$\lim_{n\rightarrow \infty }\left ( \int_{0}^{2\pi}{\frac{\cos(nx)}{x^{2}+n^{2}}} dx\right ) $$

Integral $\int_{0}^{2\pi}{\frac{\cos(nx)}{x^{2}+n^{2}}}dx $ is the main issue here.

First try to show Uniform convergence of $\frac{\cos(nx)}{x^{2}+n^{2}}$ . We can do this by Weirestrass-M test.

Theorem:- If $\{f_{n}\}$ is a sequence of Riemann integrable functions on $[a,b]$ such that $f_{n}$ converges uniformly to $f$ , then the limiting function $f$ is Riemann integrable on $[a,b]$ and $\lim_{n\to\infty}\int_{a}^{b}f_{n}(x)\,dx=\int_{a}^{b}f(x)\,dx$ . That is we can change the order of integral and limit.

We have:-

$$\sup_{x\in[0,2\pi]}|\frac{\cos(nx)}{x^{2}+n^{2}}-0|\leq \sup_{x\in[0,2\pi]} \frac{1}{x^{2}+n^{2}}\leq \frac{1}{n^{2}}$$ .

As $\frac{1}{n^{2}}\to 0$ we have $\frac{\cos(nx)}{x^{2}+n^{2}}$ is uniformly convergent to $0$.

As for each $n\in\mathbb{N}$. $\frac{\cos(nx)}{x^{2}+n^{2}}$ is Riemann integrable in $[0,2\pi]$ as each of them is continuous on this compact interval. So applying the theorem We can interchange the order of limit and integral.

Thus we have $$ \lim_{n\to\infty}\int_{0}^{2\pi}{\frac{\cos(nx)}{x^{2}+n^{2}}} dx = \int_{0}^{2\pi}\lim_{n\to\infty}\frac{\cos(nx)}{x^{2}+n^{2}}dx=\int_{0}^{2\pi} 0\,dx = 0$$