I have recently encountered a problem involving complex integration, which is as follows.
For $k\in \mathbb{R}$ and $r \in \mathbb{R},$ we have:
$$\quad\int_{-r\pi}^{r\pi}\frac{e^{ik}-1}{|k|}dk=-\log r(1+O(1)).$$
But I'm not sure how to derive this result. Could anybody give me some solution or hint of the evaluation of this complex integral? Many thanks in advance for all your helps!