`int_(-a)^(a)f(x)dx=0` if f(x) is __________ function.

Options

A. Even
B. Periodic
C. Odd
D. Neither even nor odd

Answer

To solve the question, we need to determine the nature of the function \( f(x) \) such that the integral from \(-a\) to \(a\) of \( f(x) \) is equal to zero:

\[ \int_{-a}^{a} f(x) \, dx = 0 \]

### Step-by-step Solution:

1. **Understanding the Integral**: The integral \(\int_{-a}^{a} f(x) \, dx\) represents the area under the curve of \(f(x)\) from \(-a\) to \(a\). If this integral equals zero, it implies that the area above the x-axis is exactly canceled out by the area below the x-axis.

2. **Properties of Functions**:
- **Even Function**: A function \(f(x)\) is called even if \(f(-x) = f(x)\) for all \(x\). The integral of an even function over a symmetric interval \([-a, a]\) is generally positive and does not equal zero unless \(f(x) = 0\) everywhere.
- **Odd Function**: A function \(f(x)\) is called odd if \(f(-x) = -f(x)\) for all \(x\). The integral of an odd function over a symmetric interval \([-a, a]\) is always zero because the positive area on one side of the y-axis cancels the negative area on the other side.

3. **Conclusion**: Since we need the integral \(\int_{-a}^{a} f(x) \, dx\) to equal zero, the only type of function that satisfies this condition is an odd function.

Thus, we conclude that \( f(x) \) must be an **odd function**.

### Final Answer:
The integral \(\int_{-a}^{a} f(x) \, dx = 0\) if \( f(x) \) is an **odd function**.