I was solving one problem and I don't know if I got it right. I know that using linearity of integration, integral of sum of two or more functions is just sum of integrals of each of those functions. I don't know if that can be used for closed line integrals. My example stated:
Calculate integral $$ J = \oint _{C}\left[\mathrm{e}^{1/z}\sin\left(1 \over z\right) + {1 \over \left(z + 1\right)\left(z - 2\right)\left(z + 2\right)^{2}}\right]\mathrm{d}z $$ where $C$ is circle $\left\vert z\right\vert = 3$ using Cauchy's integral formula. I'm not sure if I can use linearity of integration here such as $\displaystyle\oint _{C}\left[\mathrm{f}\left(z\right) + \mathrm{g}\left(z\right)\right]\mathrm{d}z$ is same as $\displaystyle\oint _{C}\mathrm{f}\left(z\right)\,dz + \oint _{C}\mathrm{g}\left(z\right)\,\mathrm{d}z$.
Yes you can. And for the first integral, Cauchy integral formula does not help. You should consider Laurent series and then find the residue at z=0.