Evaluate the contour integral $$ \int_C(z^2+4iz)\sin\left(\frac{3}{z+2i}\right)\,dz $$ where $C$ is the circle $|z| = 3$ oriented in the counterclockwise direction.
Any hint how to start this? Should i use cauchy integral formula or cauchy integral formula for derivatives?
Hint: $$ \left(z^2+4iz\right)\sin\left(\frac3{z+2i}\right) =\left[(z+2i)^2+4\right]\left[\frac3{z+2i}-\frac16\frac{27}{(z+2i)^3}+O\!\left(\frac1{(z+2i)^5}\right)\right] $$ and consider the residue at $z=-2i$.