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 satisfy \( f(-x) = -f(x) \) and have symmetric properties about the origin.

Answer 1

The Correct Option is B

Step 1: Use the property of definite integrals
The integral \( \int_{-a}^a f(x) \, dx = 0 \) if the function \( f(x) \) is odd. A function \( f(x) \) is odd if: \[ f(-x) = -f(x). \] 
Step 2: Simplify the integral for odd functions
For odd functions: \[ \int_{-a}^a f(x) \, dx = \int_{-a}^0 f(x) \, dx + \int_{0}^a f(x) \, dx. \] 
Since \( f(-x) = -f(x) \), the integral over \( [-a, 0] \) cancels with the integral over \( [0, a] \), resulting in: \[ \int_{-a}^a f(x) \, dx = 0. \] 
Step 3: {Conclude the result}
The given integral equals 0 when \( f(-x) = -f(x) \).