`int _(a) ^(b) f (x) dx =`

Options

A. `int _(b) ^(a) f (x) dx `
B. `- int _(a) ^(b) f (x) dx`
C. `- int _(b) ^(a) f (x) dx `
D. `int _(0) ^(a) f (x) dx`

Answer

To solve the given question, we need to evaluate the integral \( \int_{a}^{b} f(x) \, dx \) and understand the property of integration that allows us to change the limits of integration.

### Step-by-Step Solution:

1. **Understanding the Integral**: We start with the integral \( \int_{a}^{b} f(x) \, dx \). This represents the area under the curve of the function \( f(x) \) from \( x = a \) to \( x = b \).

2. **Using the Property of Integration**: There is a fundamental property of definite integrals which states that:
\[ \int_{a}^{b} f(x) \, dx = -\int_{b}^{a} f(x) \, dx \]
This means that if we reverse the limits of integration, we must also change the sign of the integral.

3. **Applying the Property**: According to this property, we can rewrite our integral:
\[ \int_{a}^{b} f(x) \, dx = -\int_{b}^{a} f(x) \, dx \]

4. **Conclusion**: Therefore, if we see the options provided in the question, the correct answer will be the expression that reflects this property, which is \( -\int_{b}^{a} f(x) \, dx \).

### Final Answer:
\[ \int_{a}^{b} f(x) \, dx = -\int_{b}^{a} f(x) \, dx \]