Find the Function g(x)=x^2+3

$g (x) = x^{2} + 3$

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.

$\int x^{2} d x + \int 3 d x$

Step 3

By the Power Rule, the integral of $x^{2}$ with respect to $x$ is $\frac{1}{3} x^{3}$ .

$\frac{1}{3} x^{3} + C + \int 3 d x$

Step 4

Apply the constant rule.

$\frac{1}{3} x^{3} + C + 3 x + C$

Step 5

Simplify.

$\frac{1}{3} x^{3} + 3 x + C$

Step 6

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

$G (x) = \frac{1}{3} x^{3} + 3 x + C$