Contour integration question

Question

$$\int_C \frac{e^z}{z(z-7)} dz$$ where $C$ is the circle of centre $0$ and radius $2$ traversed in the anticlockwise direction.

I'm having trouble integrating this, I've attempted to use Cauchy's integration formula but it's not working, can somebody please help.

Answer 1

If you know the residue theorem: The only pole enclosed is the one at 0. Then your integrals value is $$ 2\pi i(\lim_{z\rightarrow 0}\frac{e^z}{z-7})=-\frac{2\pi i}{7} $$ If you do not know the residue theorem, follow the suggestion in the comments and rewrite the integrand as $$ \frac{e^z}{7}(\frac{1}{z-7}-\frac{1}{z}) $$ using partial fraction techniques. Then apply Cauchy's integral formula.

Answer 2

Note that $f(z)=\frac{e^z}{z-7}$ is holomorphic on $|z|\le2$, thus,

$$\oint_C\frac{f(z)}z\ dz=2\pi if(0)$$

By Cauchy's integral formula.