Find the Function f(x)=-3x+5

$f (x) = - 3 x + 5$

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.

$\int - 3 x d x + \int 5 d x$

Since $- 3$ is constant with respect to $x$ , move $- 3$ out of the integral.

$- 3 \int x d x + \int 5 d x$

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

$- 3 (\frac{1}{2} x^{2} + C) + \int 5 d x$

Apply the constant rule.

$- 3 (\frac{1}{2} x^{2} + C) + 5 x + C$

Simplify.

Combine $\frac{1}{2}$ and $x^{2}$ .

$- 3 (\frac{x^{2}}{2} + C) + 5 x + C$

Simplify.

$\frac{- 3 x^{2}}{2} + 5 x + C$

Move the negative in front of the fraction.

$- \frac{3 x^{2}}{2} + 5 x + C$

Reorder terms.

$- \frac{3}{2} x^{2} + 5 x + C$

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

$F (x) = - \frac{3}{2} x^{2} + 5 x + C$