Calculus Examples

$\int_{2}^{3} x (x - 2) d x$

Step 1

Multiply $x (x - 2)$ .

$\int_{2}^{3} x \cdot x + x \cdot - 2 d x$

Step 2

Simplify.

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Step 2.1

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

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

Step 2.2

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

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

Step 2.3

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

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

Step 2.4

Add $1$ and $1$ .

$\int_{2}^{3} x^{2} + x \cdot - 2 d x$

Step 2.5

Move $- 2$ to the left of $x$ .

$\int_{2}^{3} x^{2} - 2 x d x$

Step 3

Split the single integral into multiple integrals.

$\int_{2}^{3} x^{2} d x + \int_{2}^{3} - 2 x d x$

Step 4

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

$\frac{1}{3} x^{3}]_{2}^{3} + \int_{2}^{3} - 2 x d x$

Step 5

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

$\frac{1}{3} x^{3}]_{2}^{3} - 2 \int_{2}^{3} x d x$

Step 6

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

$\frac{1}{3} x^{3}]_{2}^{3} - 2 (\frac{1}{2} x^{2}]_{2}^{3})$

Step 7

Simplify the answer.

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Step 7.1

Combine $\frac{1}{2}$ and $x^{2}$ .

$\frac{1}{3} x^{3}]_{2}^{3} - 2 (\frac{x^{2}}{2}]_{2}^{3})$

Step 7.2

Substitute and simplify.

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Step 7.2.1

Evaluate $\frac{1}{3} x^{3}$ at $3$ and at $2$ .

$(\frac{1}{3} \cdot 3^{3}) - \frac{1}{3} \cdot 2^{3} - 2 (\frac{x^{2}}{2}]_{2}^{3})$

Step 7.2.2

Evaluate $\frac{x^{2}}{2}$ at $3$ and at $2$ .

$\frac{1}{3} \cdot 3^{3} - \frac{1}{3} \cdot 2^{3} - 2 (\frac{3^{2}}{2} - \frac{2^{2}}{2})$

Step 7.2.3

Simplify.

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Step 7.2.3.1

Raise $3$ to the power of $3$ .

$\frac{1}{3} \cdot 27 - \frac{1}{3} \cdot 2^{3} - 2 (\frac{3^{2}}{2} - \frac{2^{2}}{2})$

Step 7.2.3.2

Combine $\frac{1}{3}$ and $27$ .

$\frac{27}{3} - \frac{1}{3} \cdot 2^{3} - 2 (\frac{3^{2}}{2} - \frac{2^{2}}{2})$

Step 7.2.3.3

Cancel the common factor of $27$ and $3$ .

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Step 7.2.3.3.1

Factor $3$ out of $27$ .

$\frac{3 \cdot 9}{3} - \frac{1}{3} \cdot 2^{3} - 2 (\frac{3^{2}}{2} - \frac{2^{2}}{2})$

Step 7.2.3.3.2

Cancel the common factors.

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Step 7.2.3.3.2.1

Factor $3$ out of $3$ .

$\frac{3 \cdot 9}{3 (1)} - \frac{1}{3} \cdot 2^{3} - 2 (\frac{3^{2}}{2} - \frac{2^{2}}{2})$

Step 7.2.3.3.2.2

Cancel the common factor.

$\frac{3 \cdot 9}{3 \cdot 1} - \frac{1}{3} \cdot 2^{3} - 2 (\frac{3^{2}}{2} - \frac{2^{2}}{2})$

Step 7.2.3.3.2.3

Rewrite the expression.

$\frac{9}{1} - \frac{1}{3} \cdot 2^{3} - 2 (\frac{3^{2}}{2} - \frac{2^{2}}{2})$

Step 7.2.3.3.2.4

Divide $9$ by $1$ .

$9 - \frac{1}{3} \cdot 2^{3} - 2 (\frac{3^{2}}{2} - \frac{2^{2}}{2})$

Step 7.2.3.4

Raise $2$ to the power of $3$ .

$9 - \frac{1}{3} \cdot 8 - 2 (\frac{3^{2}}{2} - \frac{2^{2}}{2})$

Step 7.2.3.5

Multiply $8$ by $- 1$ .

$9 - 8 (\frac{1}{3}) - 2 (\frac{3^{2}}{2} - \frac{2^{2}}{2})$

Step 7.2.3.6

Combine $- 8$ and $\frac{1}{3}$ .

$9 + \frac{- 8}{3} - 2 (\frac{3^{2}}{2} - \frac{2^{2}}{2})$

Step 7.2.3.7

Move the negative in front of the fraction.

$9 - \frac{8}{3} - 2 (\frac{3^{2}}{2} - \frac{2^{2}}{2})$

Step 7.2.3.8

To write $9$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$9 \cdot \frac{3}{3} - \frac{8}{3} - 2 (\frac{3^{2}}{2} - \frac{2^{2}}{2})$

Step 7.2.3.9

Combine $9$ and $\frac{3}{3}$ .

$\frac{9 \cdot 3}{3} - \frac{8}{3} - 2 (\frac{3^{2}}{2} - \frac{2^{2}}{2})$

Step 7.2.3.10

Combine the numerators over the common denominator.

$\frac{9 \cdot 3 - 8}{3} - 2 (\frac{3^{2}}{2} - \frac{2^{2}}{2})$

Step 7.2.3.11

Simplify the numerator.

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Step 7.2.3.11.1

Multiply $9$ by $3$ .

$\frac{27 - 8}{3} - 2 (\frac{3^{2}}{2} - \frac{2^{2}}{2})$

Step 7.2.3.11.2

Subtract $8$ from $27$ .

$\frac{19}{3} - 2 (\frac{3^{2}}{2} - \frac{2^{2}}{2})$

Step 7.2.3.12

Raise $3$ to the power of $2$ .

$\frac{19}{3} - 2 (\frac{9}{2} - \frac{2^{2}}{2})$

Step 7.2.3.13

Raise $2$ to the power of $2$ .

$\frac{19}{3} - 2 (\frac{9}{2} - \frac{4}{2})$

Step 7.2.3.14

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

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Step 7.2.3.14.1

Factor $2$ out of $4$ .

$\frac{19}{3} - 2 (\frac{9}{2} - \frac{2 \cdot 2}{2})$

Step 7.2.3.14.2

Cancel the common factors.

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Step 7.2.3.14.2.1

Factor $2$ out of $2$ .

$\frac{19}{3} - 2 (\frac{9}{2} - \frac{2 \cdot 2}{2 (1)})$

Step 7.2.3.14.2.2

Cancel the common factor.

$\frac{19}{3} - 2 (\frac{9}{2} - \frac{2 \cdot 2}{2 \cdot 1})$

Step 7.2.3.14.2.3

Rewrite the expression.

$\frac{19}{3} - 2 (\frac{9}{2} - \frac{2}{1})$

Step 7.2.3.14.2.4

Divide $2$ by $1$ .

$\frac{19}{3} - 2 (\frac{9}{2} - 1 \cdot 2)$

Step 7.2.3.15

Multiply $- 1$ by $2$ .

$\frac{19}{3} - 2 (\frac{9}{2} - 2)$

Step 7.2.3.16

To write $- 2$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$\frac{19}{3} - 2 (\frac{9}{2} - 2 \cdot \frac{2}{2})$

Step 7.2.3.17

Combine $- 2$ and $\frac{2}{2}$ .

$\frac{19}{3} - 2 (\frac{9}{2} + \frac{- 2 \cdot 2}{2})$

Step 7.2.3.18

Combine the numerators over the common denominator.

$\frac{19}{3} - 2 \frac{9 - 2 \cdot 2}{2}$

Step 7.2.3.19

Simplify the numerator.

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Step 7.2.3.19.1

Multiply $- 2$ by $2$ .

$\frac{19}{3} - 2 \frac{9 - 4}{2}$

Step 7.2.3.19.2

Subtract $4$ from $9$ .

$\frac{19}{3} - 2 (\frac{5}{2})$

Step 7.2.3.20

Combine $- 2$ and $\frac{5}{2}$ .

$\frac{19}{3} + \frac{- 2 \cdot 5}{2}$

Step 7.2.3.21

Multiply $- 2$ by $5$ .

$\frac{19}{3} + \frac{- 10}{2}$

Step 7.2.3.22

Cancel the common factor of $- 10$ and $2$ .

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Step 7.2.3.22.1

Factor $2$ out of $- 10$ .

$\frac{19}{3} + \frac{2 \cdot - 5}{2}$

Step 7.2.3.22.2

Cancel the common factors.

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Step 7.2.3.22.2.1

Factor $2$ out of $2$ .

$\frac{19}{3} + \frac{2 \cdot - 5}{2 (1)}$

Step 7.2.3.22.2.2

Cancel the common factor.

$\frac{19}{3} + \frac{2 \cdot - 5}{2 \cdot 1}$

Step 7.2.3.22.2.3

Rewrite the expression.

$\frac{19}{3} + \frac{- 5}{1}$

Step 7.2.3.22.2.4

Divide $- 5$ by $1$ .

$\frac{19}{3} - 5$

Step 7.2.3.23

To write $- 5$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$\frac{19}{3} - 5 \cdot \frac{3}{3}$

Step 7.2.3.24

Combine $- 5$ and $\frac{3}{3}$ .

$\frac{19}{3} + \frac{- 5 \cdot 3}{3}$

Step 7.2.3.25

Combine the numerators over the common denominator.

$\frac{19 - 5 \cdot 3}{3}$

Step 7.2.3.26

Simplify the numerator.

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Step 7.2.3.26.1

Multiply $- 5$ by $3$ .

$\frac{19 - 15}{3}$

Step 7.2.3.26.2

Subtract $15$ from $19$ .

$\frac{4}{3}$

Step 8

The result can be shown in multiple forms.

Exact Form:

$\frac{4}{3}$

Decimal Form:

$1.\overline{3}$

Mixed Number Form:

$1 \frac{1}{3}$

Step 9