I have trouble showing that $\displaystyle \int_{2}^x\frac{1}{(\log(x))^k}dx=O(\frac{x}{(\log (x))^k})$ as $\displaystyle x\to\infty$

A hint I've been given, is to split the integral in 2: $\displaystyle \int_{2}^{\sqrt{x}}+\int_{\sqrt{x}}^x$

I don't really see the point to that: I guess that: $\displaystyle \int_{\sqrt{x}}^x\to 0 $ as $\displaystyle x\to \infty$

My guess is that I need some powerseries-expansion. Any suggestion will be greatly appreciated.