$ \int\limits_C {\left( {z - z_0 } \right)^m dz} $ Evaluate the line integral $
\int\limits_C {\left( {z - z_0 } \right)^m dz} 
$ , where $C$ is the circle centered at $z_0$ with radius r>0, and:
$i)$ $m$ is an integer $m 
 \geqslant 0
$
$ii)$ m is an integer $m<0$
I want to see how can I solve this , to see some examples of integrals )=.

Parameterize the circle by $C(t) = z_0 + r e^{it} $ with $t\in (-\pi,\pi].$ Using the definition of a path integral, $$ \int_{\gamma} f(z) dz = \int^b_a f(\gamma(t)) \gamma'(t) dt$$
we have $$ \int_{C} (z-z_0)^m dz = \int^{\pi}_{-\pi} r^m e^{imt} ire^{it} dt = ir^{m+1} \int^{\pi}_{-\pi} e^{i (m+1)t} dt.$$
Now if $m+1\neq 0$ then the last integral is simply zero, seen either by direct evaluation or noting that the exponential goes through exactly $|m+1|$ periods. If $m+1=0$ then the integrand is $1$ and the value of the integral is thus $2\pi i.$
To summarize, $$\int_C (z-z_0)^m \, dz =  \left\{
     \begin{array}{lr}
       2\pi i  \text{ if }m=-1 \\
       0  \text{ if } m\in \mathbb{Z}\setminus \{-1\}
     \end{array}
   \right.$$
