Answer
Diverges
Work Step by Step
Apply a direct limit comparison test.
We have $-1 \leq \cos x$
This suggests that $\dfrac{1}{x} \leq \dfrac{2 +\cos x}{x}$ for all $x \geq \pi$
We also know:
$\lim\limits_{p \to \infty}\int_{\pi}^{p} \dfrac{dx}{x}=\lim\limits_{p \to \infty}[\ln|x|]_{\pi}^{p} =\infty$
Hence, the given integral diverges by the direct comparison test.