Calculus Examples

$\int_{- 1}^{2} (3 u - 2) (u + 1) d u$

Step 1

Simplify.

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Step 1.1

Apply the distributive property.

$\int_{- 1}^{2} 3 u (u + 1) - 2 (u + 1) d u$

Step 1.2

Apply the distributive property.

$\int_{- 1}^{2} 3 u \cdot u + 3 u \cdot 1 - 2 (u + 1) d u$

Step 1.3

Apply the distributive property.

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

Step 1.4

Move $u$ .

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

Step 1.5

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

$\int_{- 1}^{2} 3 (u^{1} u) + 3 \cdot 1 u - 2 u - 2 \cdot 1 d u$

Step 1.6

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

$\int_{- 1}^{2} 3 (u^{1} u^{1}) + 3 \cdot 1 u - 2 u - 2 \cdot 1 d u$

Step 1.7

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

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

Step 1.8

Add $1$ and $1$ .

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

Step 1.9

Multiply $3$ by $1$ .

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

Step 1.10

Multiply $- 2$ by $1$ .

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

Step 1.11

Subtract $2 u$ from $3 u$ .

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

Step 2

Split the single integral into multiple integrals.

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

Step 3

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

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

Step 4

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

$3 (\frac{1}{3} u^{3}]_{- 1}^{2}) + \int_{- 1}^{2} u d u + \int_{- 1}^{2} - 2 d u$

Step 5

Combine $\frac{1}{3}$ and $u^{3}$ .

$3 (\frac{u^{3}}{3}]_{- 1}^{2}) + \int_{- 1}^{2} u d u + \int_{- 1}^{2} - 2 d u$

Step 6

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

$3 (\frac{u^{3}}{3}]_{- 1}^{2}) + \frac{1}{2} u^{2}]_{- 1}^{2} + \int_{- 1}^{2} - 2 d u$

Step 7

Apply the constant rule.

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

Step 8

Simplify the answer.

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Step 8.1

Combine $\frac{1}{2} u^{2}]_{- 1}^{2}$ and $- 2 u]_{- 1}^{2}$ .

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

Step 8.2

Substitute and simplify.

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Step 8.2.1

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

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

Step 8.2.2

Evaluate $\frac{1}{2} u^{2} - 2 u$ at $2$ and at $- 1$ .

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

Step 8.2.3

Simplify.

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Step 8.2.3.1

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

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

Step 8.2.3.2

Raise $- 1$ to the power of $3$ .

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

Step 8.2.3.3

Move the negative in front of the fraction.

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

Step 8.2.3.4

Multiply $- 1$ by $- 1$ .

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

Step 8.2.3.5

Multiply $\frac{1}{3}$ by $1$ .

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

Step 8.2.3.6

Combine the numerators over the common denominator.

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

Step 8.2.3.7

Add $8$ and $1$ .

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

Step 8.2.3.8

Cancel the common factor of $9$ and $3$ .

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Step 8.2.3.8.1

Factor $3$ out of $9$ .

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

Step 8.2.3.8.2

Cancel the common factors.

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Step 8.2.3.8.2.1

Factor $3$ out of $3$ .

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

Step 8.2.3.8.2.2

Cancel the common factor.

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

Step 8.2.3.8.2.3

Rewrite the expression.

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

Step 8.2.3.8.2.4

Divide $3$ by $1$ .

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

Step 8.2.3.9

Multiply $3$ by $3$ .

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

Step 8.2.3.10

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

$9 + \frac{1}{2} \cdot 4 - 2 \cdot 2 - (\frac{1}{2} {(- 1)}^{2} - 2 \cdot - 1)$

Step 8.2.3.11

Combine $\frac{1}{2}$ and $4$ .

$9 + \frac{4}{2} - 2 \cdot 2 - (\frac{1}{2} {(- 1)}^{2} - 2 \cdot - 1)$

Step 8.2.3.12

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

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Step 8.2.3.12.1

Factor $2$ out of $4$ .

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

Step 8.2.3.12.2

Cancel the common factors.

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Step 8.2.3.12.2.1

Factor $2$ out of $2$ .

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

Step 8.2.3.12.2.2

Cancel the common factor.

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

Step 8.2.3.12.2.3

Rewrite the expression.

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

Step 8.2.3.12.2.4

Divide $2$ by $1$ .

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

Step 8.2.3.13

Multiply $- 2$ by $2$ .

$9 + 2 - 4 - (\frac{1}{2} {(- 1)}^{2} - 2 \cdot - 1)$

Step 8.2.3.14

Subtract $4$ from $2$ .

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

Step 8.2.3.15

Raise $- 1$ to the power of $2$ .

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

Step 8.2.3.16

Multiply $\frac{1}{2}$ by $1$ .

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

Step 8.2.3.17

Multiply $- 2$ by $- 1$ .

$9 - 2 - (\frac{1}{2} + 2)$

Step 8.2.3.18

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

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

Step 8.2.3.19

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

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

Step 8.2.3.20

Combine the numerators over the common denominator.

$9 - 2 - \frac{1 + 2 \cdot 2}{2}$

Step 8.2.3.21

Simplify the numerator.

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Step 8.2.3.21.1

Multiply $2$ by $2$ .

$9 - 2 - \frac{1 + 4}{2}$

Step 8.2.3.21.2

Add $1$ and $4$ .

$9 - 2 - \frac{5}{2}$

Step 8.2.3.22

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

$9 - 2 \cdot \frac{2}{2} - \frac{5}{2}$

Step 8.2.3.23

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

$9 + \frac{- 2 \cdot 2}{2} - \frac{5}{2}$

Step 8.2.3.24

Combine the numerators over the common denominator.

$9 + \frac{- 2 \cdot 2 - 5}{2}$

Step 8.2.3.25

Simplify the numerator.

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Step 8.2.3.25.1

Multiply $- 2$ by $2$ .

$9 + \frac{- 4 - 5}{2}$

Step 8.2.3.25.2

Subtract $5$ from $- 4$ .

$9 + \frac{- 9}{2}$

Step 8.2.3.26

Move the negative in front of the fraction.

$9 - \frac{9}{2}$

Step 8.2.3.27

To write $9$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

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

Step 8.2.3.28

Combine $9$ and $\frac{2}{2}$ .

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

Step 8.2.3.29

Combine the numerators over the common denominator.

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

Step 8.2.3.30

Simplify the numerator.

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Step 8.2.3.30.1

Multiply $9$ by $2$ .

$\frac{18 - 9}{2}$

Step 8.2.3.30.2

Subtract $9$ from $18$ .

$\frac{9}{2}$

Step 9

The result can be shown in multiple forms.

Exact Form:

$\frac{9}{2}$

Decimal Form:

$4.5$

Mixed Number Form:

$4 \frac{1}{2}$

Step 10