Integrate complex conjugate
Solution 1:
Let the curve $\gamma$ be parameterized with $z(t)=x(t)+iy(t)\in C^1$, $t\in [0,1]$. Then, we can write the integral as
$$\begin{align} \oint_\gamma \bar z\,dz &=\int_0^1 \bar z(t)\frac{dz(t)}{dt}\,dt\\\\ &=\int_0^1 (x(t)x'(t)+y(t)y'(t))\,dt+i\int_0^1 (x(t)y'(t)-y(t)x'(t))\,dt\\\\ &=\frac12 \int_0^1 (x^2(t)+y^2(t))'\,dt+\color{blue}{i\int_0^1(x(t)y(t))'\,dt}-\color{red}{i2\int_0^1 y(t)x'(t)\,dt} \\\\ &=0+\color{blue}{0}-\color{red}{i2(-\text{Area enclosed by}\,\,\gamma)}\\\\ &=i2\times (\text{Area enclosed by}\,\,\gamma) \end{align}$$