For a, b ∈ R and a lt b , ∫_a^b(f(x))/(f(x)+f(a+b-x)) dx =

The correct answer is Option (4) → \(\displaystyle\frac{b-a}{2}\)

Let

    \( \displaystyle I = \int_{a}^{b}\frac{f(x)}{f(x)+f(a+b-x)}\,dx \qquad (1) \)

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

    \( \displaystyle I = \int_{a}^{b}\frac{f(a+b-x)}{f(a+b-x)+f(x)}\,dx \qquad (2) \)

Adding (1) and (2), the numerators add up to the common denominator:

    \( \displaystyle 2I = \int_{a}^{b}1\,dx = b - a \)

    \( \displaystyle I = \frac{b-a}{2} \)

# Mathematics 07-08-2022 Slot 2: paper Q26-50 (Section B1) as new rows, part 2