Let $\gamma(t)=e^{it},t \in [0,2\pi]$. We take a look at:
$\int_\gamma \frac{\cos(z)}{z^3+2z^2} dz$
We let $f(z)=\frac{\cos(z)}{z^2+2z}$ and have
$\int_\gamma \frac{\cos(z)}{z^3+2z^2} dz=\int_\gamma \frac{f(z)}{z}dz$
The problem is that $\lim_{z\rightarrow 0} f(z)=\infty$. Now, I can't use the Cauchy integral formula to evaluate the integral. What can be done?