Find the value of integral $\int _\gamma (z^2-3|z|+Im\; z) dz$

Question

Find the value of integral $\int _\gamma (z^2-3|z|+Im\; z) dz$ Where $\gamma$ s the quarter circle centered at the origin extending from $2$ to $2i$.

Answer 1

Using $\gamma: z(t)=2e^{it}$ where $0\leq t\leq\dfrac{\pi}{2}$ we have $$\int_\gamma (z^2-3|z|+{\bf Im} z) dz=\int_0^\frac{\pi}{2}\left(e^{2it}-3+\sin t\right)ie^{it}dt=\dfrac{14}{3}-\dfrac{\pi}{2}-\dfrac{19}{3}i$$

Answer 2

Could you use $$\int _\gamma f(z) dz=\int _a^b f(z(t)) z'(t)dt$$ Here $\gamma: (2\cos t, 2 \sin t)$, $t\in[0,\frac{\pi}{ 2}]$