This is a homework problem, I need to calculate $\int\frac{e^{iz}}{z}\, dz$ over some curves, but I can find a useful parametrization of $z$ that make this calculations not so difficult.
For example I have to find the integral over the arc in the superior semi-plane of $re^{i\theta}$ from $r$ to $-r$ I would like to use a polar parametrization but I end with
$\int_0^{\pi}ie^{re^{i\theta}}\, d\theta$, and then I don't know how to proceed, also a hint of a parametrization for lines would be useful.