Let $\gamma(z_0,R)$ denote the circular contour $z_0+Re^{it}$ for $0\leq t \leq 2\pi$. Evaluate $$\int_{\gamma(0,1)}\frac{e^z+e^{-z}}{z^n}dz \hspace{10mm} n=1,2,3,\cdots .$$
Using Cauchy's formula: \begin{align*} \int_{\gamma(0,1)}\frac{e^z+e^{-z}}{z^n}dz & = 2\int_{\gamma(0,1)}\frac{e^z+e^{-z}}{2z^n}dz \\ & = 2\int_{\gamma(0,1)}\frac{\cosh}{z^n} \\ & = \frac{4\pi i}{(n-1)!}\left(\frac{d^{n-1}}{dz^{n-1}}\cosh(z) \right). \end{align*} Then evaluating at $0$ gives $$\frac{4\pi i}{(n-1)!}\sinh(0)\hspace{10mm} \text{if $n$ is even}$$ and $$\frac{4\pi i}{(n-1)!}\cosh(0) \hspace{10mm} \text{if $n$ is odd}.$$
Please let me know if I have made a mistake.