Using the Argument Principle, evaluate $$\int_{|z|=4}\frac{2z\sin(z)+z^2\cos(z)}{z^2\sin(z)}dz$$
The argument principle is the idea that $$\int_\Gamma \frac{f'(z)}{f(z)}dz=2\pi i(N-P)$$ where N is the number of zeros where N is the number of zeros of $f$ on $\Gamma$ and P is the number of poles of $f$ on $\Gamma$.
I was hoping someone could double check my work.
I think that $N=3$ and $P=0$, so $2\pi i(3-0)=6\pi i$. Is this correct? A classmate of mine got $10\pi i$, so I'm trying to get some thoughts on my solution.