For a homework exercise, we are asked to prove that $$ \int_{2}^{x} \frac{dt}{(\log(t))^{k}} = O \Big{(} \frac{x}{(\log(x))^{k}} \Big{)} \quad \text{, as } x \to \infty . $$ The following hint is given: "Split the integral into $\int_{2}^{f(x)} + \int_{f(x)}^{x} $ for a well-chosen function $f(x)$ with $ 2\leq f(x) \leq x$ and estimate both parts from above.
I tried to use the functions $f(x) = \sqrt{x}$ and $f(x) = \log(x)$, but these functions don't seem to work.
Do you know which function $f$ I ought to choose in order to be able to prove what was asked?