Calculus Examples

$3 x^{3}$

Step 1

Write $3 x^{3}$ as a function.

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

Step 2

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

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

Step 3

Set up the integral to solve.

$F (x) = \int 3 x^{3} d x$

Step 4

Since $3$ is constant with respect to $x$ , move $3$ out of the integral.

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

Step 5

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

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

Step 6

Simplify the answer.

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Step 6.1

Rewrite $3 (\frac{1}{4} x^{4} + C)$ as $3 (\frac{1}{4}) x^{4} + C$ .

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

Step 6.2

Combine $3$ and $\frac{1}{4}$ .

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

Step 7

The answer is the antiderivative of the function $f (x) = 3 x^{3}$ .

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