Find the Function f(x)=2x^3

$f (x) = 2 x^{3}$

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

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

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

Step 3

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

$2 (\frac{1}{4} x^{4} + C)$

Step 4

Simplify the answer.

Rewrite $2 (\frac{1}{4} x^{4} + C)$ as $2 (\frac{1}{4}) x^{4} + C$ .

$2 (\frac{1}{4}) x^{4} + C$

Simplify.

Combine $2$ and $\frac{1}{4}$ .

$\frac{2}{4} x^{4} + C$

Cancel the common factor of $2$ and $4$ .

Factor $2$ out of $2$ .

$\frac{2 (1)}{4} x^{4} + C$

Cancel the common factors.

Factor $2$ out of $4$ .

$\frac{2 \cdot 1}{2 \cdot 2} x^{4} + C$

Cancel the common factor.

$\frac{2 \cdot 1}{2 \cdot 2} x^{4} + C$

Rewrite the expression.

$\frac{1}{2} x^{4} + C$

Step 5

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

$F (x) = \frac{1}{2} x^{4} + C$