Question

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

• A. \( f(-x) = f(x) \)
• B. \( f(-x) = -f(x) \)
• C. \( f(a - x) = f(x) \)
• D. \( f(a - x) = -f(x) \)

Hint:

Odd functions are symmetric about the origin, and their integrals over symmetric intervals cancel out.

Answer 1

The Correct Option is B

If \( f(-x) = -f(x) \), the function is odd. For odd functions, the integral over a symmetric interval \( [-a, a] \) is: \[ \int_{-a}^a f(x) \, dx = 0. \] This is because the areas above and below the x-axis cancel out.
Final Answer: \( \boxed{f(-x) = -f(x)} \)