Find the Function f(x)=x^3+3

$f (x) = x^{3} + 3$

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.

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

Step 3

By the Power Rule, the integral of $x^{3}$ with respect to $x$ is $\frac{1}{4} x^{4}$ .

$\frac{1}{4} x^{4} + C + \int 3 d x$

Step 4

Apply the constant rule.

$\frac{1}{4} x^{4} + C + 3 x + C$

Step 5

Simplify.

$\frac{1}{4} x^{4} + 3 x + C$

Step 6

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

$F (x) = \frac{1}{4} x^{4} + 3 x + C$