Calculus Examples

$(12 x^{2} - 4 x) d x$

Step 1

Write $(12 x^{2} - 4 x) d x$ as a function.

$f (x) = (12 x^{2} - 4 x) d x$

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.

$F (x) = \int (12 x^{2} - 4 x) d x d x$

Step 4

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

$d \int (12 x^{2} - 4 x) x d x$

Step 5

Simplify.

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Step 5.1

Apply the distributive property.

$d \int 12 x^{2} x - 4 x \cdot x d x$

Step 5.2

Raise $x$ to the power of $1$ .

$d \int 12 (x^{2} x^{1}) - 4 x \cdot x d x$

Step 5.3

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$d \int 12 x^{2 + 1} - 4 x \cdot x d x$

Step 5.4

Add $2$ and $1$ .

$d \int 12 x^{3} - 4 x \cdot x d x$

Step 5.5

Raise $x$ to the power of $1$ .

$d \int 12 x^{3} - 4 (x^{1} x) d x$

Step 5.6

Raise $x$ to the power of $1$ .

$d \int 12 x^{3} - 4 (x^{1} x^{1}) d x$

Step 5.7

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$d \int 12 x^{3} - 4 x^{1 + 1} d x$

Step 5.8

Add $1$ and $1$ .

$d \int 12 x^{3} - 4 x^{2} d x$

Step 6

Split the single integral into multiple integrals.

$d (\int 12 x^{3} d x + \int - 4 x^{2} d x)$

Step 7

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

$d (12 \int x^{3} d x + \int - 4 x^{2} d x)$

Step 8

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

$d (12 (\frac{1}{4} x^{4} + C) + \int - 4 x^{2} d x)$

Step 9

Since $- 4$ is constant with respect to $x$ , move $- 4$ out of the integral.

$d (12 (\frac{1}{4} x^{4} + C) - 4 \int x^{2} d x)$

Step 10

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

$d (12 (\frac{1}{4} x^{4} + C) - 4 (\frac{1}{3} x^{3} + C))$

Step 11

Simplify.

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

Step 12

Reorder terms.

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

Step 13

The answer is the antiderivative of the function $f (x) = (12 x^{2} - 4 x) d x$ .

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