If $C_0$ denotes a positively oriented circle $|z-z_0|=R$, then $\int_{C_0}$ $(z-z_0)^{n-1} dz$ = $\left\{ \begin{array}{lr} 0 & n=\pm1, \pm2, ...\\ 2\pi i & n=0\\ \end{array} \right.$
Show that if $C$ is the boundary of the rectangle $0$ $\le$ $x$ $\le$ $3$, $0$ $\le$ $y$ $\le$ $2$, described in the positive sense, then $\int_C$$(z-2-i)^{n-1}dz$ = $\left\{ \begin{array}{lr} 0 & n=\pm1, \pm2, ...\\ 2\pi i & n=0\\ \end{array} \right.$
For the first part, write
$z=R e^{i \phi}$
for $\phi \in [0,2 \pi)$.
For the second, break $C$ into 4 pieces:
$C_1: z=x$, $x \in [0,3]$
$C_2: z=3+i y$, $y \in [0,2]$
$C_3: z=2 i + x$, $x \in [3,0]$
$C_4: z=i y$, $y \in [2,0]$
Add the 4 pieces together.