To find $f(x)$, we need to integrate the given derivative $\frac{d}{dx} f(x) = 2x + \frac{3}{x}$.

First, we integrate each term separately: $\int (2x + \frac{3}{x}) \, dx$

1. Integrate $2x$: $\int 2x \, dx = x^2 + C_1$

2. Integrate $\frac{3}{x}$: $\int \frac{3}{x} \, dx = 3 \ln |x| + C_2$

Combining these results, we get: $f(x) = x^2 + 3 \ln |x| + C$

Now, we use the initial condition $f(1) = 1$ to find the constant $C$: $f(1) = 1^2 + 3 \ln |1| + C = 1 + 0 + C = 1$ $C = 0$

Therefore, the function $f(x)$ is: $f(x) = x^2 + 3 \ln |x|$

Do you want more details or have any questions? Here are 5 related questions you can ask:

1. How do you integrate $2x$ step-by-step?
2. Why is $\ln |x|$ used instead of $\ln x$?
3. What are the properties of the natural logarithm function?
4. Can you explain how to use initial conditions to find constants in integrals?
5. What are the rules for integration of basic functions?

Tip: Always check initial conditions carefully to ensure the correct determination of constants in indefinite integrals.