f and g are continuous functions on interval [ a, b ]. Given that f ( a -x)= f (x) and g (x)+ g ( a -x)= a, show that ∫_0^a f (x) g (x) d x=a/2 ∫_0^a…

\[\begin{aligned} & I=\int_{0}^{\mathrm{a}} \mathrm{f}(x) \mathrm{g}(x) \mathrm{d} x \\ & =\int_{0}^{\mathrm{a}} \mathrm{f}(\mathrm{a}-x) \mathrm{g}(\mathrm{a}-x) \mathrm{d} x \\ & =\int_{0}^{\mathrm{a}} \mathrm{f}(x)[\mathrm{a}-\mathrm{g}(x)] \mathrm{d} x \\ & I=\mathrm{a} \int_{0}^{\mathrm{a}} \mathrm{f}(x) \mathrm{d} x-\int_{0}^{\mathrm{a}} \mathrm{f}(x) \mathrm{g}(x) \mathrm{d} x \\ & \text { Adding, we get } I=\frac{\mathrm{a}}{2} \int_{0}^{\mathrm{a}} \mathrm{f}(x) \mathrm{d} x \end{aligned} \]