If f(a-x) = f(x) , then ∫_0^a x f(x) dx is equal to

The correct answer is Option (1) → \(\displaystyle\frac{a}{2}\int_{0}^{a}\displaystyle f(x)\,dx\)

Let

    \(\displaystyle I = \int_{0}^{a} x f(x)\,dx\)

By the property \(\int_{0}^{a} g(x)\,dx = \int_{0}^{a} g(a - x)\,dx\), replace x by a − x:

    \(\displaystyle I = \int_{0}^{a} (a - x) f(a - x)\,dx\)

Since f(a − x) = f(x),

    \(\displaystyle I = \int_{0}^{a} (a - x) f(x)\,dx\)

    \(\displaystyle = a\int_{0}^{a} f(x)\,dx - \int_{0}^{a} x f(x)\,dx\)

    \(\displaystyle = a\int_{0}^{a} f(x)\,dx - I\)

So

    \(\displaystyle 2I = a\int_{0}^{a} f(x)\,dx\)

    \(\displaystyle I = \frac{a}{2}\int_{0}^{a} f(x)\,dx\)