Find the Function f(x)=5x

$f (x) = 5 x$

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 $5$ is constant with respect to $x$ , move $5$ out of the integral.

$5 \int x d x$

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

$5 (\frac{1}{2} x^{2} + C)$

Simplify the answer.

Rewrite $5 (\frac{1}{2} x^{2} + C)$ as $5 (\frac{1}{2}) x^{2} + C$ .

$5 (\frac{1}{2}) x^{2} + C$

Combine $5$ and $\frac{1}{2}$ .

$\frac{5}{2} x^{2} + C$

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

$F (x) = \frac{5}{2} x^{2} + C$