Finding the path over which to compute a complex line integral when converted from a real integral

Question

I have the integral $$\int_0^{2\pi}\frac{1}{5+3\cos t}dt.$$ And I want to convert this to a complex line integral.

My idea was to use $\cos t=(e^{it}+e^{-it})/2$, but what is the path $a$ over which I integrate? And does the integral become $\int_a\frac{1}{5+3/2(e^{it}+e^{-it})}dt$?

Answer 1

Make the change of variable $z = e^{it} $, then: $$\cos t = \frac {z+z^{-1}}{2} \quad dt = \frac {dz}{iz}$$ and the path is the unit circle.