Use Cauchy integral formula to determine an integral

Question

I am trying to use CIF to solve $$\int_\gamma (z^2-4)^{-1} dz$$ where $\gamma$ is the unit circle traversed once in the positive direction.

If I let $f(z) = z^2-4$, then $f$ is not analytic at $\pm2$. However, neither of these points lie within $\gamma$. Thus, I am not sure how to continue this problem.

Answer 1

$(z^2 - 4)^{-1}$ is holomorphic within the unit disk. Thus the integral is $0$ by Goursat's theorem.