Find the Function f(x)=g(x)

$f (x) = g (x)$

The function $F (x)$ can be found by evaluating the indefinite integral of the derivative $f (x)$ .

$F (x) = \int f (x) d x$

Since $g$ is constant with respect to $x$ , move $g$ out of the integral.

$g \int x d x$

By the Power Rule, the integral of $x$ with respect to $x$ is $\frac{1}{2} x^{2}$ .

$g (\frac{1}{2} x^{2} + C)$

Simplify the answer.

Rewrite $g (\frac{1}{2} x^{2} + C)$ as $g \frac{1}{2} x^{2} + C$ .

$g \frac{1}{2} x^{2} + C$

Simplify.

Combine $g$ and $\frac{1}{2}$ .

$\frac{g}{2} x^{2} + C$

Combine $\frac{g}{2}$ and $x^{2}$ .

$\frac{g x^{2}}{2} + C$

$\frac{1}{2} g x^{2} + C$

The function $F$ if derived from the integral of the derivative of the function. This is valid by the fundamental theorem of calculus.

$F (x) = \frac{1}{2} \cdot (g x^{2}) + C$