If `f'(x)=x+(1)/(x)" and " f(1)=(5)/(2)," then " f(x)=`

Options

A. `(x^(2))/(2)+log|x|+2`
B. `(x^(2))/(2)+log|x|-2`
C. `(x^(2))/(2)+log|x|+4`
D. `(x^(2))/(2)+log|x|-4`

Answer

To solve the problem, we need to find the function \( f(x) \) given that its derivative \( f'(x) = x + \frac{1}{x} \) and that \( f(1) = \frac{5}{2} \).

### Step-by-Step Solution:

1. **Start with the given derivative**:
\[ f'(x) = x + \frac{1}{x} \]

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

3. **Perform the integration**:
- The integral of \( x \) is \( \frac{x^2}{2} \).
- The integral of \( \frac{1}{x} \) is \( \log |x| \).
- Therefore, we have:
\[ f(x) = \frac{x^2}{2} + \log |x| + C \]
where \( C \) is the constant of integration.

4. **Use the initial condition \( f(1) = \frac{5}{2} \) to find \( C \)**:
\[ f(1) = \frac{1^2}{2} + \log(1) + C \]
\[ f(1) = \frac{1}{2} + 0 + C = \frac{5}{2} \]
\[ C = \frac{5}{2} - \frac{1}{2} = \frac{4}{2} = 2 \]

5. **Substitute \( C \) back into the equation for \( f(x) \)**:
\[ f(x) = \frac{x^2}{2} + \log |x| + 2 \]

6. **Final result**:
\[ f(x) = \frac{x^2}{2} + \log x + 2 \]

### Final Answer:
\[ f(x) = \frac{x^2}{2} + \log x + 2 \]