If `f'(x)=x+(1)/(x),` then the value of f(x) is

Options

A. `x^(2)+log x+C`
B. `(x^(2))/(2)+log|x|+C`
C. `(x)/(2)+logx+C`
D. None of these

Explanation

To find the function \( f(x) \) given that \( f'(x) = x + \frac{1}{x} \), we need to integrate \( f'(x) \).

### Step-by-Step Solution:

1. **Write down the derivative**:
\[ f'(x) = x + \frac{1}{x} \]

2. **Integrate \( f'(x) \)**:
\[ f(x) = \int f'(x) \, dx = \int \left( x + \frac{1}{x} \right) \, dx \]

3. **Split the integral**:
\[ f(x) = \int x \, dx + \int \frac{1}{x} \, dx \]

4. **Calculate the first integral**:
\[ \int x \, dx = \frac{x^2}{2} \]

5. **Calculate the second integral**:
\[ \int \frac{1}{x} \, dx = \log |x| \]

6. **Combine the results**:
\[ f(x) = \frac{x^2}{2} + \log |x| + C \]
where \( C \) is the constant of integration.

### Final Answer:
Thus, the value of \( f(x) \) is:
\[ f(x) = \frac{x^2}{2} + \log |x| + C \]