I'm having trouble integrating the following integral
$$\int_0^{2\pi} \frac{d\theta}{3 -2\cos\theta + \sin\theta}$$
I make the substitutions $z= e^{i\theta}$ , $\sin\theta = \frac{1}{2i}(z-\frac{1}{z})$ and $\cos\theta = \frac{1}{2}(z+\frac{1}{z})$, and then calculate the roots of the denominator.
I calculate the two roots to be
$z_+ = \frac{2-i}{5}$
$z_- = 2-i$
The problem is that $z_+$ lies on the unit circle itself, and therefore I can't apply Residue theorem. I think I somehow have to calculate the Principal value, but I'm not sure how?