Question

If $ \int_0^{2a} f(x) \, dx = 2 \int_0^a f(x) \, dx, \text{ then} $

• A. \( f(2a - x) = -f(x) \)
• B. \( f(2a - x) = f(x) \)
• C. \( f(x) \) is an odd function
• D. \( f(x) \) is an even function

Hint:

Use the properties of definite integrals and symmetry to simplify the evaluation of integrals, especially when the limits of integration are symmetric.

Answer 1

The Correct Option is A

We are given the condition: \[ \int_0^{2a} f(x) \, dx = 2 \int_0^a f(x) \, dx \]
Step 1: Use the property of definite integrals
The integral from 0 to \( 2a \) can be split into two integrals: \[ \int_0^{2a} f(x) \, dx = \int_0^a f(x) \, dx + \int_a^{2a} f(x) \, dx \]
Step 2: Set up the equation
Substitute the given condition into this equation: \[ 2 \int_0^a f(x) \, dx = \int_0^a f(x) \, dx + \int_a^{2a} f(x) \, dx \] Simplifying this gives: \[ \int_a^{2a} f(x) \, dx = \int_0^a f(x) \, dx \]
Step 3: Conclusion
Thus, \( f(2a - x) = -f(x) \), which means \( f(x) \) is an odd function.
Hence, the correct answer is option (a).