If $\frac{d}{dx}f(x) = 2x + \frac{3}{x}$ and $f(1) = 1$, then $f(x)$ is

$x^2 + 3 \log |x|$

The correct answer is Option (2) → $x^2 + 3 \log |x|$ ##

The given differential equation is

$\frac{d}{dx}f(x) = 2x + \frac{3}{x}$

Integrating both sides:

$\int d[f(x)] = \int \left(2x + \frac{3}{x}\right) dx$

$f(x) = \frac{2x^2}{2} + 3 \log |x| + c$

$f(x) = x^2 + 3 \log |x| + c$

Putting $x = 1, f(1) = 1$ in above equation:

$1 = 1 + 3 \log |1| + c$

$c = 0$

$∴f(x) = x^2 + 3 \log |x|$