Complex integration of $\int_{C} (z^2+3z) dz$ counter clock wise from $(2,0)$ to $(0,2)$ along the curve C:$|z|=2$.

Question

Using the parametrization: $$z=2e^{it}$$ and $t\in[0,\pi/2]$ and $$z^{'}(t)=2ie^{it}$$ I have solved the integral to get: $$\frac{-44}{3}-\frac{8i}{3}$$ Need to know if my answer is correct as I am not able to verify from original source

Answer 1

Using Cauchy: $$ \int_C z^2 + 3z \, dz = \int_2^0 t^2 + 3t \, dt - \int_0^2 3t + i t^2 \, dt = -6\int_0^2 t \,dt - \int_0^2 t^2 \,dt -i \int_0^2 t^2 \, dt = -12 - \frac 83 - i \frac 83 = -\frac {44} 3 -i \frac 83. $$

Answer 2

$f(z) = z^2+3z$ is the derivative of $F(z) = \frac 13 z^3 + \frac 32 z^2$. Therefore $$ \int_C f(z) \, dz = F(2i) - F(2) = \left(-\frac 83 i-6 \right) - \left(\frac 83 + 6\right) = -\frac{44}{3} -\frac 83 i $$ for any curve connecting $2$ with $2i$.