I have a problem where I must evaluate
$$\int_{|z| = 1} \dfrac{z^2}{\sqrt[4]{P(z)}} dz$$
Where $P(z)$ is a polynomial with degree at least four and has exactly four roots in the unit circle. I know I am supposed to use the Cauchy-Goursat theorem and construct a contour that avoids the discontinuities but I cannot imagine how to draw this contour or how to evaluate it.
Can I just draw any contour that avoids the singularities?
I would also appreciate any examples of contour integrals which include a fourth root in the denominator.