Evaluate the complex integral: $\int_{|z|=1}xdz$

Question

$\int_{|z|=1}xdz$

I ended up with $2\pi$ as my final answer, can anyone confirm and/or give me a shorter way to do it? Mine involved lots of sines & cosines.

Answer 1

$$\int_C x dz=\int_C \frac{z+\overline z}{2} dz\\ \left(z=e^{i\theta}, dz=izd\theta\right)\\ =\int_0^{2\pi}\frac{e^{i\theta}+e^{-i\theta}}{2}ie^{i\theta}d\theta =\int_0^{2\pi}i\frac{e^{2i\theta}+1}{2}d\theta\\ =\left[\frac{e^{2i\theta}}{4}+i\frac{\theta}{2}\right]_0^{2\pi}=\pi i $$