\( \int_{-a}^a f(x) \, dx = 0 \), if:
Show Hint
- \( f(-x) = f(x) \)
- \( f(-x) = -f(x) \)
- \( f(a - x) = f(x) \)
- \( f(a - x) = -f(x) \)
The Correct Option is B
Solution and Explanation
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 \( f(-x) = -f(x) \).
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