Methods to find the value of $\int_0^{2\pi} \frac{dt}{z+e^{it}}$ Given $z \in \mathbb{C}$, with $|z|\neq1$, what are the classic methods to determine the value of:
$$\int_0^{2\pi} \dfrac{dt}{z+e^{it}}$$
I thought of rewriting the integral with only real integrands but it seems very tedious. I am not asking necessarily for an answer but I would like to know how can I approach this kind of integral in order to be quite efficient.
I never saw contour integration, so I am not going to use it in any case. But if you have a solution that involves this concept, I would be glad to read it anyway.

Consider the mean value theorem for harmonic functions:
$$f\left( z \right)=\frac{1}{2\pi }\int\limits_{0}^{2\pi }{f\left( z+{{e}^{it}} \right)dt}\Rightarrow \int\limits_{0}^{2\pi }{\frac{1}{z+{{e}^{it}}}dt}=\frac{2\pi }{z}$$
This is true for all z where f is analytic in a unit circle centred at z. For example this is true for all z with $\operatorname{Re}\left( z \right)>1$ etc.
