Evaluate this complex line integral

Question

I think I'm close but there's a part I'm not sure of. I have to find the value of $$\int|z|^2(\bar z + i)dz$$ and $\gamma$ is the circle given by $|z - i| = 1$ with counterclockwise orientation. I already figured out that $z(t) = cos(t) + i(1 + sin(t))$ so $|z|^2 = 2 + 2sin(t)$ and $\bar z = cos(t) - i(1 + sin(t))$. However when I multiply by $z'(t)$, I get $-sin(t) + icos(t)$ but my teacher said the integrand is not supposed to be analytic. Can someone see if I made a mistake somewhere?

Answer 1

In your notation, $\overline z+i=\cos t-i\sin t$ and $dz=i(\cos t+i\sin t)\,dt$. Then $$(\overline z+i)\,dz=i(\cos t-i\sin t)(\cos t+i\sin t)\,dt=i\,dt$$ so your integral is $$\int_0^{2\pi}i(2+2\sin t)\,dt$$ etc.