I want to solve the line integral. $$f\left( z\right) =\dfrac{1}{e^{z}+1}$$ $$\int _{C}\dfrac{1}{e^{z}+1}dz$$ $C$ is a circle with a radius of 1 centered on 1.
I think this.
$$e^{z}+1=0$$ So $z=3i\pi$ is isolated singularity. $$z=z\left( \theta \right) =-1+e^{i\theta }$$ $$\int _{0}^{2\pi }\dfrac{ie^{i\theta }}{e^{-1+e^{i\theta}}+1}=\int _{0}^{2\pi }\dfrac{ie^{i\theta +e^{i}\theta }}{e^{e^{i}\theta }+e}d\theta$$ I don't know what to do next. Please tell me. By the way, the answer is enter link description here