Calculus Examples

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

Step 1

Multiply $12 x^{2} (x - 1)$ .

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

Step 2

Simplify.

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Step 2.1

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

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

Step 2.2

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

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

Step 2.3

Add $1$ and $2$ .

$\int 12 x^{3} + 12 x^{2} \cdot - 1 d x$

Step 2.4

Multiply $- 1$ by $12$ .

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

Step 3

Split the single integral into multiple integrals.

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

Step 4

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

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

Step 5

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

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

Step 6

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

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

Step 7

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

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

Step 8

Simplify.

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Step 8.1

Simplify.

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

Step 8.2

Simplify.

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Step 8.2.1

Combine $12$ and $\frac{1}{4}$ .

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

Step 8.2.2

Cancel the common factor of $12$ and $4$ .

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Step 8.2.2.1

Factor $4$ out of $12$ .

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

Step 8.2.2.2

Cancel the common factors.

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Step 8.2.2.2.1

Factor $4$ out of $4$ .

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

Step 8.2.2.2.2

Cancel the common factor.

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

Step 8.2.2.2.3

Rewrite the expression.

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

Step 8.2.2.2.4

Divide $3$ by $1$ .

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

Step 8.2.3

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

$3 x^{4} + \frac{- 12}{3} x^{3} + C$

Step 8.2.4

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

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Step 8.2.4.1

Factor $3$ out of $- 12$ .

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

Step 8.2.4.2

Cancel the common factors.

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Step 8.2.4.2.1

Factor $3$ out of $3$ .

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

Step 8.2.4.2.2

Cancel the common factor.

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

Step 8.2.4.2.3

Rewrite the expression.

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

Step 8.2.4.2.4

Divide $- 4$ by $1$ .

$3 x^{4} - 4 x^{3} + C$