Find the Function f(x)=-2/3x+5

$f (x) = - \frac{2}{3} x + 5$

Step 1

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

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

Step 2

Split the single integral into multiple integrals.

$\int - \frac{2}{3} x d x + \int 5 d x$

Step 3

Since $- \frac{2}{3}$ is constant with respect to $x$ , move $- \frac{2}{3}$ out of the integral.

$- \frac{2}{3} \int x d x + \int 5 d x$

Step 4

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

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

Step 5

Apply the constant rule.

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

Step 6

Simplify.

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

Simplify.

Multiply $\frac{1}{2}$ by $\frac{2}{3}$ .

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

Multiply $2$ by $3$ .

$- \frac{2}{6} x^{2} + 5 x + C$

Cancel the common factor of $2$ and $6$ .

Factor $2$ out of $2$ .

$- \frac{2 (1)}{6} x^{2} + 5 x + C$

Cancel the common factors.

Factor $2$ out of $6$ .

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

Cancel the common factor.

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

Rewrite the expression.

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

Step 7

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

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