Let $f(z) = z +\overline{z}$. Compute the integral $$\int_{C}f(z)dz$$ where $C$ is a line segment from $0$ to $1 + i.$
Attempt:
Let $\phi(t)=t+ti$ for $t\in[0,1]$ be the paramerization of $C$.
Since $z +\overline{z}=2\Re(z)$,
$$\int_{C}f(z)dz=\int_{0}^{1}f(\phi(t))\phi^{'}(t)dt=\int_{0}^{1}2t(1+i)dt=2(1+i)\int_{0}^{1}tdt=1+i.$$
Is my attempt correct? Thanks!