Integrate $f(x) =\operatorname{Im} z$ counterclockwise around the unit circle.

Question

Integrate $f(x) = \operatorname{Im}z$ counterclockwise around the unit circle. I understand that Cauchy's Formula does not apply here. I don't quite know how to separate the imaginary from the real for $z=e^{it}$ and then converting this into the integral $\int f(z(t))z'(t)\,dt$.

Answer 1

You compute$$\int_0^{2\pi}\operatorname{Im}(e^{it})ie^{it}\,\mathrm dt=i\int_0^{2\pi}\sin(t)\cos(t)\,\mathrm dt-\int_0^{2\pi}\sin^2(t)\,\mathrm dt.$$Can you take it from here?