Answer
See the proof.
Work Step by Step
$$\int_{a}^{b}(cx+d)f(x)dx=c\int_{a}^{b}xf(x)dx+d\int_{a}^{b}f(x)dx=c\overline{x}\cdot A+d\int_{a}^{b}f(x)dx$$
Since $A$ is the area of the region bounded between the curve of $f$, the $x$-axis on $[a,b]$ then:
$$A=\int_{a}^{b}f(x)dx$$
so:
$$\int_{a}^{b}(cx+d)f(x)dx=c\int_{a}^{b}xf(x)dx+d\int_{a}^{b}f(x)dx=c\overline{x}\cdot \int_{a}^{b}f(x)dx+d\int_{a}^{b}f(x)dx=(c\overline{x}+d)\int_{a}^{b}f(x)dx$$
