Let f and g be continuous functions on [0,a] such that f(x)=f(a-x) and g(x)+g(a-x)=4, then…

Question

Let $f$ and $g$ be continuous functions on $[0,a]$ such that $f(x)=f(a-x)$ and $g(x)+g(a-x)=4$, then $\displaystyle\int_0^af(x)g(x)\,dx$ is equal to:

Accepted Answer

Let $I=\displaystyle\int_0^af(x)g(x)\,dx$. Using $x\to a-x$:
$$I=\int_0^af(a-x)g(a-x)\,dx=\int_0^af(x)[4-g(x)]\,dx=4\int_0^af(x)dx-I$$

$$\Rightarrow2I=4\int_0^af(x)dx\Rightarrow I=2\int_0^af(x)\,dx$$

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