Find the Function f(x)=5-2x

$f (x) = 5 - 2 x$

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.

$\int 5 d x + \int - 2 x d x$

Apply the constant rule.

$5 x + C + \int - 2 x d x$

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

$5 x + C - 2 \int x d x$

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

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

Simplify.

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

Simplify.

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

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

Cancel the common factor of $- 2$ and $2$ .

Factor $2$ out of $- 2$ .

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

Cancel the common factors.

Factor $2$ out of $2$ .

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

Cancel the common factor.

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

Rewrite the expression.

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

Divide $- 1$ by $1$ .

$5 x - x^{2} + C$

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

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