If `(d)/(dx)f(x)=g(x)` for `a le x le b` then, `int_(a)^(b) f(x) g(x) dx` equals

Options

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

Answer

To solve the problem, we need to evaluate the integral \( \int_{a}^{b} f(x) g(x) \, dx \) given that \( \frac{d}{dx} f(x) = g(x) \).

### Step-by-Step Solution:

1. **Understand the Given Information**:
We know that \( g(x) = \frac{d}{dx} f(x) \). This means that \( g(x) \) is the derivative of \( f(x) \).

2. **Use Integration by Parts**:
We can use integration by parts, which states:
\[ \int u \, dv = uv - \int v \, du \]
Here, we can let:
- \( u = f(x) \) (hence \( du = g(x) \, dx \))
- \( dv = g(x) \, dx \) (which means \( v = f(x) \))

3. **Apply Integration by Parts**:
Applying integration by parts, 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. **Evaluate the Boundary Terms**:
The boundary terms become:
\[ \left[ f(x) f(x) \right]_{a}^{b} = f(b) f(b) - f(a) f(a) = f(b)^2 - f(a)^2 \]

5. **Combine the Results**:
We can rearrange our equation:
\[ \int_{a}^{b} f(x) g(x) \, dx + \int_{a}^{b} f(x) g(x) \, dx = f(b)^2 - f(a)^2 \]
This simplifies to:
\[ 2 \int_{a}^{b} f(x) g(x) \, dx = f(b)^2 - f(a)^2 \]

6. **Solve for the Integral**:
Dividing both sides by 2 gives us:
\[ \int_{a}^{b} f(x) g(x) \, dx = \frac{1}{2} (f(b)^2 - f(a)^2) \]

### Final Result:
Thus, we conclude that:
\[ \int_{a}^{b} f(x) g(x) \, dx = \frac{1}{2} (f(b)^2 - f(a)^2) \]