$\displaystyle \int_{0}^{2\pi} \frac{d\theta}{37 - 12cos\theta} = \oint_{C} \frac{dz/iz}{37 - 12(\frac{z}{2} + \frac{1}{2z})}$

$\displaystyle = \oint_{C} \frac{dz/iz}{37 - 6z - \frac{6}{z}}$

$\displaystyle = \oint_{C} \frac{dz}{i(37z - 6z^{2} - 6)}$

$\displaystyle = \oint_{C} \frac{dz}{-i(6z^{2} - 37z + 6)}$

$\displaystyle = \frac{-1}{i} \oint_{C} \frac{dz}{(6z - 1)(z - 6)}$

$\displaystyle z_{1} = \frac{1}{6}, z_{2} = 6$

This is the part which I'm not really sure. I'm guessing the $\displaystyle z_{1}$ is outside because it's less than unity, but I start having trouble when it comes to identifying more complicated poles. What is the right way?

$\displaystyle Res = \left|\frac{1}{6z - 1} \right|_{z = 6} = \frac{1}{35}$

$\displaystyle \Rightarrow 2\pii(\frac{-1}{i})(\frac{1}{35}) = \frac{-2\pi}{35}$

The answer is not a negative, so it may be a careless mistake, but I've pored through the equations without spotting an error.

PS: Is \\ the tag for line break? I can't seem to get it to work.