If `int f(x) dx = 2 cos sqrt(x) + c`, then f(x) =

Options

A. `sin sqrt(x)`
B. `-(sin sqrt(x))/(sqrt(x))`
C. `2 cos sqrt(x)`
D. None of these

Answer

To solve the problem, we need to find the function \( f(x) \) given that the integral of \( f(x) \) with respect to \( x \) is equal to \( 2 \cos(\sqrt{x}) + c \).

### Step-by-Step Solution:

1. **Start with the given integral**:
\[ \int f(x) \, dx = 2 \cos(\sqrt{x}) + c \]

2. **Differentiate both sides with respect to \( x \)**:
To find \( f(x) \), we need to differentiate the right-hand side:
\[ f(x) = \frac{d}{dx}(2 \cos(\sqrt{x}) + c) \]

3. **Apply the chain rule**:
The derivative of \( 2 \cos(\sqrt{x}) \) can be calculated using the chain rule. Let \( u = \sqrt{x} \), then:
\[ \frac{d}{dx} (2 \cos(u)) = -2 \sin(u) \cdot \frac{du}{dx} \]
where \( \frac{du}{dx} = \frac{1}{2\sqrt{x}} \).

4. **Substituting back**:
Now substituting back \( u = \sqrt{x} \):
\[ f(x) = -2 \sin(\sqrt{x}) \cdot \frac{1}{2\sqrt{x}} = -\frac{\sin(\sqrt{x})}{\sqrt{x}} \]

5. **Final result**:
Thus, we find:
\[ f(x) = -\frac{\sin(\sqrt{x})}{\sqrt{x}} \]

### Conclusion:
The function \( f(x) \) is:
\[ f(x) = -\frac{\sin(\sqrt{x})}{\sqrt{x}} \]