If $\frac{d}{dx}[f(x)] = ax + b$, and $f(0) = 0$, then $f(x)$ is equal to:

$\frac{ax^2}{2} + bx$

The correct answer is Option (2) → $\frac{ax^2}{2} + bx$ ##

The given differential equation is:

$\frac{d}{dx}[f(x)] = ax + b$

Integrating both sides:

$\int d[f(x)] = \int(ax+b)dx$

$f(x) = \frac{ax^2}{2} + bx + c$

Putting $x = 0, f(0) = 0$ in above equation:

$f(x) = \frac{ax^2}{2} + bx + c$

$0 = 0 + 0 + c$

$c = 0$

$∴f(x) = \frac{ax^2}{2} + bx$