If \(f(x)\) and \(g(x)\) are continuous functions in \([0, a]\) such that \(f(x) = f(a-x)\) and \(g(x) + g(a-x) = a\) then \(\int_{0}^{a}\displaystyle f(x)g(x)\,dx =\)
Answer & explanation
Correct answer: option 3
The correct answer is Option (3) → \({\displaystyle\frac{a}{2}}\int_{0}^{a}\displaystyle f(x)\,dx\)
Let
\( \displaystyle I = \int_{0}^{a} f(x)g(x)\,dx \)
By the property \(\int_{0}^{a} h(x)\,dx = \int_{0}^{a} h(a - x)\,dx\), and using \(f(a - x) = f(x)\):
\( \displaystyle I = \int_{0}^{a} f(a-x)\,g(a-x)\,dx = \int_{0}^{a} f(x)\,g(a-x)\,dx \)
Adding the two forms of I and using \(g(x) + g(a-x) = a\):
\( \displaystyle 2I = \int_{0}^{a} f(x)\left[g(x) + g(a-x)\right]dx \)
\( \displaystyle = a\int_{0}^{a} f(x)\,dx \)
\( \displaystyle I = \frac{a}{2}\int_{0}^{a} f(x)\,dx \)