Find the Function f(x)=2x^2

$f (x) = 2 x^{2}$

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 $2$ is constant with respect to $x$ , move $2$ out of the integral.

$2 \int x^{2} d x$

By the Power Rule, the integral of $x^{2}$ with respect to $x$ is $\frac{1}{3} x^{3}$ .

$2 (\frac{1}{3} x^{3} + C)$

Simplify the answer.

Rewrite $2 (\frac{1}{3} x^{3} + C)$ as $2 (\frac{1}{3}) x^{3} + C$ .

$2 (\frac{1}{3}) x^{3} + C$

Combine $2$ and $\frac{1}{3}$ .

$\frac{2}{3} x^{3} + C$

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

$F (x) = \frac{2}{3} x^{3} + C$