Help with complex valued integral

Question

I'm given that $f(z)=z \bar{z}=x^{2}+y^2 $ with $z=x+iy$ and $r(t)= <cost,sint>,\quad 0 \leq t \leq 2\pi$.

I was able to evaluate the line integral by parametrization of f using r(t).

I am stuck trying to evaluate $\int z \bar{z} dz$ over the unit circle and checking for analyticity.

Answer 1

The function $f$ is not analytic. Actually, it is differentiable at $0$ and only there, as you can check from the Cauchy-Riemann equations.

And that integral is equal to$$\int_0^{2\pi}f(\cos\theta+i\sin\theta)(-\sin\theta+i\cos\theta)\,\mathrm d\theta=\int_o^{2\pi}-\sin\theta+i\cos\theta\,\mathrm d\theta.$$Can you take it from here?