Find the Function f(x)=5-x

$f (x) = 5 - x$

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

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

Split the single integral into multiple integrals.

$\int 5 d x + \int - x d x$

Apply the constant rule.

$5 x + C + \int - x d x$

Since $- 1$ is constant with respect to $x$ , move $- 1$ out of the integral.

$5 x + C - \int x d x$

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

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

Simplify.

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

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

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