Evaluate $\int_{C} (z-z^2)dz$ where $C$ is the (i)Upper half of the circle $|z|=1$

Question

Evaluate $\int_{C} (z-z^2)dz$ where $C$ is the

(i)Upper half of the circle $|z|=1$

(ii)Lower half of the circle.

Where $z$ is a complex number.

How can I approach this problem?

What I know:

$z=x+iy$

$\int_{C} (x+iy-(x+iy)^2)d(x+iy)$

$\int_{C} (x+iy-(x^2-y^2+2xyi))d(x+iy)$

Answer 1

Hint: $\dfrac{z^2}2-\dfrac{z^3}3$ is a primitive of the function that you want to integrate.