Evaluate $$\oint_{|z|=2}\frac{\cos(z)}{z^5 -32} dz$$
I know that there is a singularity for $z=2$. And so, if $z_{0}=2$ is the singularity, I can apply the formula
$$\oint_{\gamma} f(z) dz=2\pi i \mathrm{Res}(f,z_{0})$$
We have that $g(z)=\frac{\cos(z)}{z^4 +2z^3 +4z^2 +8z+16}$ is holomorphic and so have a Taylor series; $\frac{1}{z -2}$ is Laurent series with a simple pole with coefficient $1$
so $\mathrm{Res}(f,z_{0})=\frac{\cos(2)}{80}$ and we finally have that
$$\oint_{|z|=2}\frac{\cos(z)}{z^5 -32}dz=\pi i\frac{\cos(2)}{40}$$
Is it right?