TITLE: About problem in complex integrals
QUESTION [5 upvotes]: I solved this problem in complex integrals.
Is my answer a correct ? 
Here $z$ is a complex value:
$$
C:|z-1|=1 \ \ \ \ \ \mbox{integral path}
$$
$$
\int_C\ \frac{2z^2-5z+1}{z-1}\ dz
$$
My answer
$$
z=1+e^{i\theta} \ \ \  \ \frac{dz}{d\theta}=ie^{i\theta}
$$
$$
\int_{0}^{2\pi}\ \frac{-e^{i\theta}+2e^{2i\theta}-2}{e^{i\theta}} \cdot\ ie^{i\theta} d\theta
$$
$$
=\left[ -e^{i\theta}+ e^{2i\theta} -2i\theta \right]^{2\pi}_0=-4\pi i
$$

REPLY [4 votes]: If you are allowed to use the Cauchy's integral formula, then $$\int_{C} \frac{2 z^2 - 5x + 1}{z-1} dz = {2\pi i} \big( 2 z^2 - 5z + 1)_{z=1} = -4 \pi i,$$
showing that you did a great job.