How to apply cauchy integral formula.

Question

I am aware of how to solve these kind of problems using Cauchy Residue Theorem.

I Know Cauchy integral formula as a statement. But I dont know how to apply it to solve these kind of problems, because the problem specifically asking to use Cauchy Integral formula.

Kindly tell me how to apply Cauchy Integral Formula.

Answer 1

Let $f(z)=\frac{z+2}{z-2}$. Then, $f$ is holomorphic on $|z-i|=2$. By Cauchy's Integral Formula, $$ f'(-1)=\frac{1}{2\pi i}\int_{C}\frac{f(z)}{(z-(-1))^{2}}dz, $$ since $-1$ lies in the circle $C$. Hence, $$ \int_{C}\frac{z+2}{(z+1)^{2}(z-2)}dz=2\pi i\cdot f'(-1). $$