Find the Function g(x)=2x+9

$g (x) = 2 x + 9$

Step 1

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

$G (x) = \int g (x) d x$

Step 2

Split the single integral into multiple integrals.

$\int 2 x d x + \int 9 d x$

Step 3

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

$2 \int x d x + \int 9 d x$

Step 4

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

$2 (\frac{1}{2} x^{2} + C) + \int 9 d x$

Step 5

Apply the constant rule.

$2 (\frac{1}{2} x^{2} + C) + 9 x + C$

Step 6

Simplify.

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

$2 (\frac{x^{2}}{2} + C) + 9 x + C$

Simplify.

$x^{2} + 9 x + C$

Step 7

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

$G (x) = x^{2} + 9 x + C$