Let $f(x)$ be a continuous function on $[0,a]$ where $a>0$ and $f(x) + f(a-x) \neq 0$ on $[0,a]$. Evaluate the integral $$\int_0^a\dfrac{f(x)}{f(x)+f(a-x)}\,dx.$$
I am having trouble with this problem. So far, I have got that $$\int_0^a \dfrac{f(x)}{f(x)+f(a-x)} = \int_0^a \dfrac{f(a-x)}{f(x)+f(a-x)},$$ but I don't know how to relate these two integrals in a way that will help me figure out the problem. Any help would really be appreciated!
Converting my comment into an answer, note that you have already proven the following (Which is the tougher part of this problem): $$I=\int_0^a \frac{f(x)}{f(x)+f(a-x)}~dx=\int_0^a \frac{f(a-x)}{f(x)+f(a-x)}~dx$$ Therefore, it follows that: $$\begin{align}2I&=\int_0^a \frac{f(x)}{f(x)+f(a-x)}~dx+\int_0^a \frac{f(a-x)}{f(x)+f(a-x)}~dx\\&=\int_0^a \frac{f(x)+f(a-x)}{f(x)+f(a-x)}~dx\\&=\int_0^a 1~dx \end{align}$$ Where we have used the linearity of the integral. I suppose you know how to continue.