Let $f in C^1(0,infty)$ be an increasing function with $f(0)>0$, suppose $int_0^infty frac{1}{f(x)+f'(x)} < infty$, prove that $int_0^infty frac{1}{f(x)} < infty$.

I find it weird since the behaviour of $f’$ is random, so I don’t know how to control $f$ in terms of $f+f’$.