Evaluation of Real-Valued Integrals (Complex Analysis) How to get calculate the integration of follwing:
$$\int_{0}^{2\pi} \frac{dt}{a + cos t} (a>1)$$
My attempt:
let, $z=e^{it}$
$\implies dt = \frac{dz}{it}$
and 
$$\cos t = \frac{z + \frac{1}{z}}{2}$$
On substituting everything in the integral I got:
$$\frac{2}{i}\int_{c} \frac{dz}{z^2+2az+1}$$
Now how do I decompose this fraction so that I can use the Residue Theorem? Or is there anyother way to solve this??
Thanks for the help.

HINT: The denominator: $z^2+2az+1$ can be easily factorize using quadratic formula as follows $$z^2+2az+1=(z+a-\sqrt{a^2-1})(z+a+\sqrt{a^2-1})$$
hence, $$\frac{1}{z^2+2az+1}=\frac{A}{z+a-\sqrt{a^2-1}}+\frac{B}{z+a+\sqrt{a^2-1}}$$
$$=\frac{1}{2\sqrt{a^2-1}}\left(\frac{1}{z+a-\sqrt{a^2-1}}-\frac{1}{z+a+\sqrt{a^2-1}}\right)$$
