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) → $\frac{ax^2}{2} + bx$

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

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

Integral both side:

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

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

at $x=0$,

$f(x)=0$, $⇒0 = \frac{a(0)^2 }{2}+ b(0) + c$

$0 = c$

$∴f(x) = \frac{ax^2}{2} + bx$