Calculus Examples

$\int_{- 2}^{2} (2 x + 5) - (x^{2} + 2 x + 1) d x$

Step 1

Split the single integral into multiple integrals.

$\int_{- 2}^{2} 2 x d x + \int_{- 2}^{2} 5 d x + \int_{- 2}^{2} - (x^{2} + 2 x + 1) d x$

Step 2

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

$2 \int_{- 2}^{2} x d x + \int_{- 2}^{2} 5 d x + \int_{- 2}^{2} - (x^{2} + 2 x + 1) d x$

Step 3

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

$2 (\frac{1}{2} x^{2}]_{- 2}^{2}) + \int_{- 2}^{2} 5 d x + \int_{- 2}^{2} - (x^{2} + 2 x + 1) d x$

Step 4

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

$2 (\frac{x^{2}}{2}]_{- 2}^{2}) + \int_{- 2}^{2} 5 d x + \int_{- 2}^{2} - (x^{2} + 2 x + 1) d x$

Step 5

Apply the constant rule.

$2 (\frac{x^{2}}{2}]_{- 2}^{2}) + 5 x]_{- 2}^{2} + \int_{- 2}^{2} - (x^{2} + 2 x + 1) d x$

Step 6

Since $- 1$ is constant with respect to $x$ , move $- 1$ out of the integral.

$2 (\frac{x^{2}}{2}]_{- 2}^{2}) + 5 x]_{- 2}^{2} - \int_{- 2}^{2} x^{2} + 2 x + 1 d x$

Step 7

Split the single integral into multiple integrals.

$2 (\frac{x^{2}}{2}]_{- 2}^{2}) + 5 x]_{- 2}^{2} - (\int_{- 2}^{2} x^{2} d x + \int_{- 2}^{2} 2 x d x + \int_{- 2}^{2} d x)$

Step 8

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

$2 (\frac{x^{2}}{2}]_{- 2}^{2}) + 5 x]_{- 2}^{2} - (\frac{1}{3} x^{3}]_{- 2}^{2} + \int_{- 2}^{2} 2 x d x + \int_{- 2}^{2} d x)$

Step 9

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

$2 (\frac{x^{2}}{2}]_{- 2}^{2}) + 5 x]_{- 2}^{2} - (\frac{x^{3}}{3}]_{- 2}^{2} + \int_{- 2}^{2} 2 x d x + \int_{- 2}^{2} d x)$

Step 10

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

$2 (\frac{x^{2}}{2}]_{- 2}^{2}) + 5 x]_{- 2}^{2} - (\frac{x^{3}}{3}]_{- 2}^{2} + 2 \int_{- 2}^{2} x d x + \int_{- 2}^{2} d x)$

Step 11

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

$2 (\frac{x^{2}}{2}]_{- 2}^{2}) + 5 x]_{- 2}^{2} - (\frac{x^{3}}{3}]_{- 2}^{2} + 2 (\frac{1}{2} x^{2}]_{- 2}^{2}) + \int_{- 2}^{2} d x)$

Step 12

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

$2 (\frac{x^{2}}{2}]_{- 2}^{2}) + 5 x]_{- 2}^{2} - (\frac{x^{3}}{3}]_{- 2}^{2} + 2 (\frac{x^{2}}{2}]_{- 2}^{2}) + \int_{- 2}^{2} d x)$

Step 13

Apply the constant rule.

$2 (\frac{x^{2}}{2}]_{- 2}^{2}) + 5 x]_{- 2}^{2} - (\frac{x^{3}}{3}]_{- 2}^{2} + 2 (\frac{x^{2}}{2}]_{- 2}^{2}) + x]_{- 2}^{2})$

Step 14

Simplify the answer.

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Step 14.1

Combine $\frac{x^{3}}{3}]_{- 2}^{2}$ and $x]_{- 2}^{2}$ .

$2 (\frac{x^{2}}{2}]_{- 2}^{2}) + 5 x]_{- 2}^{2} - (2 (\frac{x^{2}}{2}]_{- 2}^{2}) + \frac{x^{3}}{3} + x]_{- 2}^{2})$

Step 14.2

Substitute and simplify.

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Step 14.2.1

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

$2 ((\frac{2^{2}}{2}) - \frac{{(- 2)}^{2}}{2}) + 5 x]_{- 2}^{2} - (2 (\frac{x^{2}}{2}]_{- 2}^{2}) + \frac{x^{3}}{3} + x]_{- 2}^{2})$

Step 14.2.2

Evaluate $5 x$ at $2$ and at $- 2$ .

$2 (\frac{2^{2}}{2} - \frac{{(- 2)}^{2}}{2}) + 5 \cdot 2 - 5 \cdot - 2 - (2 (\frac{x^{2}}{2}]_{- 2}^{2}) + \frac{x^{3}}{3} + x]_{- 2}^{2})$

Step 14.2.3

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

$2 (\frac{2^{2}}{2} - \frac{{(- 2)}^{2}}{2}) + 5 \cdot 2 - 5 \cdot - 2 - (2 ((\frac{2^{2}}{2}) - \frac{{(- 2)}^{2}}{2}) + \frac{x^{3}}{3} + x]_{- 2}^{2})$

Step 14.2.4

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

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

Step 14.2.5

Simplify.

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Step 14.2.5.1

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

$2 (\frac{4}{2} - \frac{{(- 2)}^{2}}{2}) + 5 \cdot 2 - 5 \cdot - 2 - (2 ((\frac{2^{2}}{2}) - \frac{{(- 2)}^{2}}{2}) + (\frac{2^{3}}{3} + 2) - (\frac{{(- 2)}^{3}}{3} - 2))$

Step 14.2.5.2

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

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Step 14.2.5.2.1

Factor $2$ out of $4$ .

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

Step 14.2.5.2.2

Cancel the common factors.

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Step 14.2.5.2.2.1

Factor $2$ out of $2$ .

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

Step 14.2.5.2.2.2

Cancel the common factor.

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

Step 14.2.5.2.2.3

Rewrite the expression.

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

Step 14.2.5.2.2.4

Divide $2$ by $1$ .

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

Step 14.2.5.3

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

$2 (2 - \frac{4}{2}) + 5 \cdot 2 - 5 \cdot - 2 - (2 ((\frac{2^{2}}{2}) - \frac{{(- 2)}^{2}}{2}) + (\frac{2^{3}}{3} + 2) - (\frac{{(- 2)}^{3}}{3} - 2))$

Step 14.2.5.4

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

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Step 14.2.5.4.1

Factor $2$ out of $4$ .

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

Step 14.2.5.4.2

Cancel the common factors.

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Step 14.2.5.4.2.1

Factor $2$ out of $2$ .

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

Step 14.2.5.4.2.2

Cancel the common factor.

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

Step 14.2.5.4.2.3

Rewrite the expression.

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

Step 14.2.5.4.2.4

Divide $2$ by $1$ .

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

Step 14.2.5.5

Multiply $- 1$ by $2$ .

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

Step 14.2.5.6

Subtract $2$ from $2$ .

$2 \cdot 0 + 5 \cdot 2 - 5 \cdot - 2 - (2 ((\frac{2^{2}}{2}) - \frac{{(- 2)}^{2}}{2}) + (\frac{2^{3}}{3} + 2) - (\frac{{(- 2)}^{3}}{3} - 2))$

Step 14.2.5.7

Multiply $2$ by $0$ .

$0 + 5 \cdot 2 - 5 \cdot - 2 - (2 ((\frac{2^{2}}{2}) - \frac{{(- 2)}^{2}}{2}) + (\frac{2^{3}}{3} + 2) - (\frac{{(- 2)}^{3}}{3} - 2))$

Step 14.2.5.8

Multiply $5$ by $2$ .

$0 + 10 - 5 \cdot - 2 - (2 ((\frac{2^{2}}{2}) - \frac{{(- 2)}^{2}}{2}) + (\frac{2^{3}}{3} + 2) - (\frac{{(- 2)}^{3}}{3} - 2))$

Step 14.2.5.9

Multiply $- 5$ by $- 2$ .

$0 + 10 + 10 - (2 ((\frac{2^{2}}{2}) - \frac{{(- 2)}^{2}}{2}) + (\frac{2^{3}}{3} + 2) - (\frac{{(- 2)}^{3}}{3} - 2))$

Step 14.2.5.10

Add $10$ and $10$ .

$0 + 20 - (2 ((\frac{2^{2}}{2}) - \frac{{(- 2)}^{2}}{2}) + (\frac{2^{3}}{3} + 2) - (\frac{{(- 2)}^{3}}{3} - 2))$

Step 14.2.5.11

Add $0$ and $20$ .

$20 - (2 ((\frac{2^{2}}{2}) - \frac{{(- 2)}^{2}}{2}) + (\frac{2^{3}}{3} + 2) - (\frac{{(- 2)}^{3}}{3} - 2))$

Step 14.2.5.12

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

$20 - (2 (\frac{4}{2} - \frac{{(- 2)}^{2}}{2}) + (\frac{2^{3}}{3} + 2) - (\frac{{(- 2)}^{3}}{3} - 2))$

Step 14.2.5.13

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

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Step 14.2.5.13.1

Factor $2$ out of $4$ .

$20 - (2 (\frac{2 \cdot 2}{2} - \frac{{(- 2)}^{2}}{2}) + (\frac{2^{3}}{3} + 2) - (\frac{{(- 2)}^{3}}{3} - 2))$

Step 14.2.5.13.2

Cancel the common factors.

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Step 14.2.5.13.2.1

Factor $2$ out of $2$ .

$20 - (2 (\frac{2 \cdot 2}{2 (1)} - \frac{{(- 2)}^{2}}{2}) + (\frac{2^{3}}{3} + 2) - (\frac{{(- 2)}^{3}}{3} - 2))$

Step 14.2.5.13.2.2

Cancel the common factor.

$20 - (2 (\frac{2 \cdot 2}{2 \cdot 1} - \frac{{(- 2)}^{2}}{2}) + (\frac{2^{3}}{3} + 2) - (\frac{{(- 2)}^{3}}{3} - 2))$

Step 14.2.5.13.2.3

Rewrite the expression.

$20 - (2 (\frac{2}{1} - \frac{{(- 2)}^{2}}{2}) + (\frac{2^{3}}{3} + 2) - (\frac{{(- 2)}^{3}}{3} - 2))$

Step 14.2.5.13.2.4

Divide $2$ by $1$ .

$20 - (2 (2 - \frac{{(- 2)}^{2}}{2}) + (\frac{2^{3}}{3} + 2) - (\frac{{(- 2)}^{3}}{3} - 2))$

Step 14.2.5.14

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

$20 - (2 (2 - \frac{4}{2}) + (\frac{2^{3}}{3} + 2) - (\frac{{(- 2)}^{3}}{3} - 2))$

Step 14.2.5.15

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

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Step 14.2.5.15.1

Factor $2$ out of $4$ .

$20 - (2 (2 - \frac{2 \cdot 2}{2}) + (\frac{2^{3}}{3} + 2) - (\frac{{(- 2)}^{3}}{3} - 2))$

Step 14.2.5.15.2

Cancel the common factors.

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Step 14.2.5.15.2.1

Factor $2$ out of $2$ .

$20 - (2 (2 - \frac{2 \cdot 2}{2 (1)}) + (\frac{2^{3}}{3} + 2) - (\frac{{(- 2)}^{3}}{3} - 2))$

Step 14.2.5.15.2.2

Cancel the common factor.

$20 - (2 (2 - \frac{2 \cdot 2}{2 \cdot 1}) + (\frac{2^{3}}{3} + 2) - (\frac{{(- 2)}^{3}}{3} - 2))$

Step 14.2.5.15.2.3

Rewrite the expression.

$20 - (2 (2 - \frac{2}{1}) + (\frac{2^{3}}{3} + 2) - (\frac{{(- 2)}^{3}}{3} - 2))$

Step 14.2.5.15.2.4

Divide $2$ by $1$ .

$20 - (2 (2 - 1 \cdot 2) + (\frac{2^{3}}{3} + 2) - (\frac{{(- 2)}^{3}}{3} - 2))$

Step 14.2.5.16

Multiply $- 1$ by $2$ .

$20 - (2 (2 - 2) + (\frac{2^{3}}{3} + 2) - (\frac{{(- 2)}^{3}}{3} - 2))$

Step 14.2.5.17

Subtract $2$ from $2$ .

$20 - (2 \cdot 0 + (\frac{2^{3}}{3} + 2) - (\frac{{(- 2)}^{3}}{3} - 2))$

Step 14.2.5.18

Multiply $2$ by $0$ .

$20 - (0 + (\frac{2^{3}}{3} + 2) - (\frac{{(- 2)}^{3}}{3} - 2))$

Step 14.2.5.19

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

$20 - (0 + \frac{8}{3} + 2 - (\frac{{(- 2)}^{3}}{3} - 2))$

Step 14.2.5.20

To write $2$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$20 - (0 + \frac{8}{3} + 2 \cdot \frac{3}{3} - (\frac{{(- 2)}^{3}}{3} - 2))$

Step 14.2.5.21

Combine $2$ and $\frac{3}{3}$ .

$20 - (0 + \frac{8}{3} + \frac{2 \cdot 3}{3} - (\frac{{(- 2)}^{3}}{3} - 2))$

Step 14.2.5.22

Combine the numerators over the common denominator.

$20 - (0 + \frac{8 + 2 \cdot 3}{3} - (\frac{{(- 2)}^{3}}{3} - 2))$

Step 14.2.5.23

Simplify the numerator.

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Step 14.2.5.23.1

Multiply $2$ by $3$ .

$20 - (0 + \frac{8 + 6}{3} - (\frac{{(- 2)}^{3}}{3} - 2))$

Step 14.2.5.23.2

Add $8$ and $6$ .

$20 - (0 + \frac{14}{3} - (\frac{{(- 2)}^{3}}{3} - 2))$

Step 14.2.5.24

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

$20 - (0 + \frac{14}{3} - (\frac{- 8}{3} - 2))$

Step 14.2.5.25

Move the negative in front of the fraction.

$20 - (0 + \frac{14}{3} - (- \frac{8}{3} - 2))$

Step 14.2.5.26

To write $- 2$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$20 - (0 + \frac{14}{3} - (- \frac{8}{3} - 2 \cdot \frac{3}{3}))$

Step 14.2.5.27

Combine $- 2$ and $\frac{3}{3}$ .

$20 - (0 + \frac{14}{3} - (- \frac{8}{3} + \frac{- 2 \cdot 3}{3}))$

Step 14.2.5.28

Combine the numerators over the common denominator.

$20 - (0 + \frac{14}{3} - \frac{- 8 - 2 \cdot 3}{3})$

Step 14.2.5.29

Simplify the numerator.

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Step 14.2.5.29.1

Multiply $- 2$ by $3$ .

$20 - (0 + \frac{14}{3} - \frac{- 8 - 6}{3})$

Step 14.2.5.29.2

Subtract $6$ from $- 8$ .

$20 - (0 + \frac{14}{3} - \frac{- 14}{3})$

Step 14.2.5.30

Move the negative in front of the fraction.

$20 - (0 + \frac{14}{3} - - \frac{14}{3})$

Step 14.2.5.31

Multiply $- 1$ by $- 1$ .

$20 - (0 + \frac{14}{3} + 1 (\frac{14}{3}))$

Step 14.2.5.32

Multiply $\frac{14}{3}$ by $1$ .

$20 - (0 + \frac{14}{3} + \frac{14}{3})$

Step 14.2.5.33

Combine the numerators over the common denominator.

$20 - (0 + \frac{14 + 14}{3})$

Step 14.2.5.34

Add $14$ and $14$ .

$20 - (0 + \frac{28}{3})$

Step 14.2.5.35

Add $0$ and $\frac{28}{3}$ .

$20 - \frac{28}{3}$

Step 14.2.5.36

To write $20$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$20 \cdot \frac{3}{3} - \frac{28}{3}$

Step 14.2.5.37

Combine $20$ and $\frac{3}{3}$ .

$\frac{20 \cdot 3}{3} - \frac{28}{3}$

Step 14.2.5.38

Combine the numerators over the common denominator.

$\frac{20 \cdot 3 - 28}{3}$

Step 14.2.5.39

Simplify the numerator.

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Step 14.2.5.39.1

Multiply $20$ by $3$ .

$\frac{60 - 28}{3}$

Step 14.2.5.39.2

Subtract $28$ from $60$ .

$\frac{32}{3}$

Step 15

The result can be shown in multiple forms.

Exact Form:

$\frac{32}{3}$

Decimal Form:

$10.\overline{6}$

Mixed Number Form:

$10 \frac{2}{3}$

Step 16