Question

If \( f(a + b - x) = f(x), \) then \( \int_a^b f(x) \, dx \) is

• A. \( \frac{a + b}{2} \int_a^b f(b - x) \, dx \)
• B. \( \frac{a + b}{2} \int_a^b f(x) \, dx \)
• C. \( b - a \int_a^b f(x) \, dx \)
• D. None of these

Hint:

Use symmetry properties of the function when integrating over symmetric intervals to simplify calculations.

Answer 1

The Correct Option is B

Given the functional condition \( f(a + b - x) = f(x) \), we use symmetry in the integral to show that the value is \( \frac{a + b}{2} \int_a^b f(x) \, dx \).
Step 2: Conclusion.
The correct answer is (B), \( \frac{a + b}{2} \int_a^b f(x) \, dx \).