The problem is to compute $\int_{\gamma}e^{iz}/z$ for $\gamma:[0,\pi]\rightarrow \mathbb{C}$ by $\gamma(t)=re^{it}$ with $r>0$.

I tried to make it a closed arc, say add two segments in real line connecting $-r$ and $-\epsilon$, $\epsilon$ and $r$, then connect $-\epsilon$ and $\epsilon$ by an upper semi circle. So the integral over this contour is 0. But it turns out that $\int_{-r}^{-\epsilon}+\int_{\epsilon}^r \frac{e^{it}}{t} dt$ is not computable? Is there any other way to compute it?