I have no idea how to prove the following improper integral is diverging. The integral is $$\int_{\pi/2}^{\infty}\left |\frac{\cos(x)}{x+\sin(x)} \right |\,dx $$
It was a part of a question to prove that the same integral without the absolute sign is converging but not absolutely converging.
There's a hint in the question saying we should prove this by turning the integral into a sum of integrals each with length $\pi$.
I tried it and it leads nowhere. Any help will be appreciated.