Complex definite integral $\int_{0}^{\pi}\frac{ire^{it}}{2-2ire^{it}}dt$

Question

I am trying to evaluate the integral $$\int\limits_{0}^{\pi}\dfrac{ire^{it}}{2-2ire^{it}}dt$$ but I don't know how to proceed.

Answer 1

The substitution $z=re^{it}$ allows you to realize the problem:$$-\frac{1}{2}\int_{\gamma} \frac{z}{z+i}\mathrm{d}z$$where $\gamma$ is the upper (counterclockwise oriented) half circle with radius $r$. You can close the path on the real axis adding the segment $[-r,r]$ and finally you can apply the residue theorem. You should discuss various cases depending on $r\in[0,\infty)$.