If `(d)/(dx)f(x)=g(x)` for `a le x le b` then, `overset(b)underset(a)int f(x) g(x) dx` equals

Options

A. `g(b)-f(a)`
B. `g(b)-g(a)`
C. `(f^(2)(b)-f^(2)(a))/(2)`
D. `(g^(2)(b)-g^(2)(a))/(2)`

Answer

To solve the problem, we start with the given information that the derivative of \( f(x) \) with respect to \( x \) is \( g(x) \) for \( a \leq x \leq b \). We want to evaluate the integral

\[ \int_a^b f(x) g(x) \, dx. \]

### Step-by-Step Solution:

1. **Substitution**: We know that \( g(x) = \frac{d}{dx} f(x) \). Therefore, we can rewrite the integral as:

\[ \int_a^b f(x) g(x) \, dx = \int_a^b f(x) \frac{d}{dx} f(x) \, dx. \]

2. **Integration by Parts**: We will use integration by parts. Let:
- \( u = f(x) \) (which implies \( du = g(x) \, dx \))
- \( dv = g(x) \, dx \) (which implies \( v = f(x) \))

The formula for integration by parts is:

\[ \int u \, dv = uv - \int v \, du. \]

3. **Applying Integration by Parts**: Applying the integration by parts formula, we have:

\[ \int_a^b f(x) g(x) \, dx = \left[ f(x) f(x) \right]_a^b - \int_a^b f(x) g(x) \, dx. \]

4. **Evaluating the Boundary Terms**: The boundary terms evaluate to:

\[ \left[ f(x) f(x) \right]_a^b = f(b)f(b) - f(a)f(a) = f(b)^2 - f(a)^2. \]

5. **Final Expression**: Therefore, we can write:

\[ \int_a^b f(x) g(x) \, dx = \frac{1}{2} \left( f(b)^2 - f(a)^2 \right). \]

6. **Conclusion**: Thus, the integral \( \int_a^b f(x) g(x) \, dx \) equals:

\[ \frac{f(b)^2 - f(a)^2}{2}. \]

### Final Answer:
The final result is:

\[ \int_a^b f(x) g(x) \, dx = \frac{f(b)^2 - f(a)^2}{2}. \]