Calculus Examples

$4 \cdot 3 x^{2}$

Step 1

Write $4 \cdot 3 x^{2}$ as a function.

$f (x) = 4 \cdot 3 x^{2}$

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.

$F (x) = \int 4 \cdot (3 x^{2}) d x$

Step 4

Multiply $3$ by $4$ .

$\int 12 x^{2} d x$

Step 5

Since $12$ is constant with respect to $x$ , move $12$ out of the integral.

$12 \int x^{2} d x$

Step 6

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

$12 (\frac{1}{3} x^{3} + C)$

Step 7

Simplify the answer.

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Step 7.1

Rewrite $12 (\frac{1}{3} x^{3} + C)$ as $12 (\frac{1}{3}) x^{3} + C$ .

$12 (\frac{1}{3}) x^{3} + C$

Step 7.2

Simplify.

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Step 7.2.1

Combine $12$ and $\frac{1}{3}$ .

$\frac{12}{3} x^{3} + C$

Step 7.2.2

Cancel the common factor of $12$ and $3$ .

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Step 7.2.2.1

Factor $3$ out of $12$ .

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

Step 7.2.2.2

Cancel the common factors.

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Step 7.2.2.2.1

Factor $3$ out of $3$ .

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

Step 7.2.2.2.2

Cancel the common factor.

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

Step 7.2.2.2.3

Rewrite the expression.

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

Step 7.2.2.2.4

Divide $4$ by $1$ .

$4 x^{3} + C$

Step 8

The answer is the antiderivative of the function $f (x) = 4 \cdot (3 x^{2})$ .

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