For $z_0\in{\Bbb C}$ such that $z_0\neq0$ and $R<|z_0|$, how can I compute $ \int_{0}^{2\pi}\frac{Re^{it}}{z_0+Re^{it}}dt? $
With help of Cauchy Theorem, one can conclude that $ \int_{0}^{2\pi}\frac{iRe^{it}}{z_0+Re^{it}}dt=\int_{\gamma}\frac{1}{z}dz=0 $ where $\gamma=\{z\in{\Bbb C}:|z-z_0|=R\}$. How can I do this without Cauchy Theorem?