I am going through a Complex Analysis text and I am trying to solve the question
A force field is given by $F=3z+5$. Find the work done in moving an object in this force fiel dalong the parabola $z=t^2+it$ from $z=0$ to $z=4+2i$.
In the solution, the author writes
$$ \begin{aligned} \text { Total work done } &=\int_{C} F \cdot d z=\operatorname{Re} \int_{C} \bar{F} \cdot d z=\operatorname{Re}\left\{\int_{C}(3 \bar{z}+5) d z\right\} \\ &=\operatorname{Re}\left\{3 \int_{C} \bar{z} d z+5 \int_{C} d z\right\}=\operatorname{Re}\left\{3\left(10-\frac{1}{2} i\right)+5(4+2 i)\right\}=50 \end{aligned} $$
Why is it that $$\int_{C} F \cdot d z=\operatorname{Re} \int_{C} \bar{F} \cdot d z$$?
This step feels pulled out of nowhere to me. Thank you!