The integral ${\int }_0^a \frac{g\left(x\right)}{f\left(x\right)+f(a-x)}dx$  vanishes, if 

Let $I={\int }_0^a \frac{g\left(x\right)}{f\left(x\right)+f(a-x)}dx$ . Then 

$I={\int }_0^a \frac{g(a-x)}{f\left(x\right)+f(a-x)}$       $\left[\begin{array}{c}\therefore {\int }_0^a \varphi \left(x\right)dx={\int }_0^a \varphi (a-x)dx\end{array}\right]$

$\therefore I+I={\int }_0^a \frac{g\left(x\right)+g(a-x)}{f\left(x\right)+f(a-x)}dx$

$\Rightarrow I=\frac{1}{2}{\int }_0^a \frac{g\left(x\right)+g(a-x)}{f\left(x\right)+f(a-x)}dx$

Clearly, I vanishes if $g(a-x)=-g\left(x\right)$