Let $f$ and $g$ be continuous functions on $[0, \mathrm{a}]$ such that $\mathrm{f}(x)=\mathrm{f}(\mathrm{a}-x)$ and $g(x)+g(a-x)=4$, then $\int_0^a f(x) g(x) d x$ is equal to

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Aicharya · Accepted Answer

Given, $\begin{aligned} & \mathrm{f}(x)=\mathrm{f}(\mathrm{a}-\mathrm{x}) \ & \mathrm{g}(x)+\mathrm{g}(\mathrm{a}-x)=4 \ & 	ext { Let } \mathrm{I}=\int_0^{\mathrm{a}} \mathrm{f}(x) \mathrm{g}(x) \mathrm{d} x \ & =\int_0^a f(a-x) \cdot g(a-x) d x \ & =\int_0^{\mathrm{a}} \mathrm{f}(x) \cdot[(4-\mathrm{g}(x)] \mathrm{d} x \ & =4 \int_0^a \mathrm{f}(x) \mathrm{d} x-\int_0^a \mathrm{f}(x) \mathrm{g}(x) \mathrm{d} x \ & \mathrm{I}=4 \int_0^{\mathrm{a}} \mathrm{f}(x) \mathrm{d} x-\mathrm{I} \ & 2 \mathrm{I}=4 \int_0^{\mathrm{a}} \mathrm{f}(x) \mathrm{d} x \ & 	herefore \quad \mathrm{I}=2 \int_0^{\mathrm{a}} \mathrm{f}(x) \mathrm{d} x \end{aligned}$

Let $f$ and $g$ be continuous functions on $[0, \mathrm{a}]$ such that…

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