Calculus Examples

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

Step 1

Remove parentheses.

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

Step 2

Split the single integral into multiple integrals.

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

Step 3

Split up the integral depending on where $x - 1$ is positive and negative.

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

Step 4

Split the single integral into multiple integrals.

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

Step 5

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

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

Step 6

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

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

Step 7

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

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

Step 8

Apply the constant rule.

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

Step 9

Split the single integral into multiple integrals.

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

Step 10

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

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

Step 11

Apply the constant rule.

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

Step 12

Apply the constant rule.

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

Step 13

Simplify the answer.

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Step 13.1

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

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

Step 13.2

Substitute and simplify.

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Step 13.2.1

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

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

Step 13.2.2

Evaluate $x$ at $1$ and at $- 2$ .

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

Step 13.2.3

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

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

Step 13.2.4

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

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

Step 13.2.5

Simplify.

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Step 13.2.5.1

One to any power is one.

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

Step 13.2.5.2

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

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

Step 13.2.5.3

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

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Step 13.2.5.3.1

Factor $2$ out of $4$ .

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

Step 13.2.5.3.2

Cancel the common factors.

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Step 13.2.5.3.2.1

Factor $2$ out of $2$ .

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

Step 13.2.5.3.2.2

Cancel the common factor.

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

Step 13.2.5.3.2.3

Rewrite the expression.

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

Step 13.2.5.3.2.4

Divide $2$ by $1$ .

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

Step 13.2.5.4

Multiply $- 1$ by $2$ .

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

Step 13.2.5.5

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

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

Step 13.2.5.6

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

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

Step 13.2.5.7

Combine the numerators over the common denominator.

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

Step 13.2.5.8

Simplify the numerator.

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Step 13.2.5.8.1

Multiply $- 2$ by $2$ .

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

Step 13.2.5.8.2

Subtract $4$ from $1$ .

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

Step 13.2.5.9

Move the negative in front of the fraction.

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

Step 13.2.5.10

Multiply $- 1$ by $- 1$ .

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

Step 13.2.5.11

Multiply $\frac{3}{2}$ by $1$ .

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

Step 13.2.5.12

Add $1$ and $2$ .

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

Step 13.2.5.13

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

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

Step 13.2.5.14

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

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

Step 13.2.5.15

Combine the numerators over the common denominator.

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

Step 13.2.5.16

Simplify the numerator.

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Step 13.2.5.16.1

Multiply $3$ by $2$ .

$\frac{3 + 6}{2} + (\frac{1}{2} \cdot 2^{2} - 1 \cdot 2) - (\frac{1}{2} \cdot 1^{2} - 1 \cdot 1) + (2 \cdot 2) - 2 \cdot - 2$

Step 13.2.5.16.2

Add $3$ and $6$ .

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

Step 13.2.5.17

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

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

Step 13.2.5.18

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

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

Step 13.2.5.19

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

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Step 13.2.5.19.1

Factor $2$ out of $4$ .

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

Step 13.2.5.19.2

Cancel the common factors.

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Step 13.2.5.19.2.1

Factor $2$ out of $2$ .

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

Step 13.2.5.19.2.2

Cancel the common factor.

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

Step 13.2.5.19.2.3

Rewrite the expression.

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

Step 13.2.5.19.2.4

Divide $2$ by $1$ .

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

Step 13.2.5.20

Multiply $- 1$ by $2$ .

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

Step 13.2.5.21

Subtract $2$ from $2$ .

$\frac{9}{2} + 0 - (\frac{1}{2} \cdot 1^{2} - 1 \cdot 1) + (2 \cdot 2) - 2 \cdot - 2$

Step 13.2.5.22

One to any power is one.

$\frac{9}{2} + 0 - (\frac{1}{2} \cdot 1 - 1 \cdot 1) + (2 \cdot 2) - 2 \cdot - 2$

Step 13.2.5.23

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

$\frac{9}{2} + 0 - (\frac{1}{2} - 1 \cdot 1) + (2 \cdot 2) - 2 \cdot - 2$

Step 13.2.5.24

Multiply $- 1$ by $1$ .

$\frac{9}{2} + 0 - (\frac{1}{2} - 1) + (2 \cdot 2) - 2 \cdot - 2$

Step 13.2.5.25

To write $- 1$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$\frac{9}{2} + 0 - (\frac{1}{2} - 1 \cdot \frac{2}{2}) + (2 \cdot 2) - 2 \cdot - 2$

Step 13.2.5.26

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

$\frac{9}{2} + 0 - (\frac{1}{2} + \frac{- 1 \cdot 2}{2}) + (2 \cdot 2) - 2 \cdot - 2$

Step 13.2.5.27

Combine the numerators over the common denominator.

$\frac{9}{2} + 0 - \frac{1 - 1 \cdot 2}{2} + (2 \cdot 2) - 2 \cdot - 2$

Step 13.2.5.28

Simplify the numerator.

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Step 13.2.5.28.1

Multiply $- 1$ by $2$ .

$\frac{9}{2} + 0 - \frac{1 - 2}{2} + (2 \cdot 2) - 2 \cdot - 2$

Step 13.2.5.28.2

Subtract $2$ from $1$ .

$\frac{9}{2} + 0 - \frac{- 1}{2} + (2 \cdot 2) - 2 \cdot - 2$

Step 13.2.5.29

Move the negative in front of the fraction.

$\frac{9}{2} + 0 - - \frac{1}{2} + (2 \cdot 2) - 2 \cdot - 2$

Step 13.2.5.30

Multiply $- 1$ by $- 1$ .

$\frac{9}{2} + 0 + 1 (\frac{1}{2}) + (2 \cdot 2) - 2 \cdot - 2$

Step 13.2.5.31

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

$\frac{9}{2} + 0 + \frac{1}{2} + (2 \cdot 2) - 2 \cdot - 2$

Step 13.2.5.32

Add $0$ and $\frac{1}{2}$ .

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

Step 13.2.5.33

Combine the numerators over the common denominator.

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

Step 13.2.5.34

Add $9$ and $1$ .

$\frac{10}{2} + (2 \cdot 2) - 2 \cdot - 2$

Step 13.2.5.35

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

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Step 13.2.5.35.1

Factor $2$ out of $10$ .

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

Step 13.2.5.35.2

Cancel the common factors.

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Step 13.2.5.35.2.1

Factor $2$ out of $2$ .

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

Step 13.2.5.35.2.2

Cancel the common factor.

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

Step 13.2.5.35.2.3

Rewrite the expression.

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

Step 13.2.5.35.2.4

Divide $5$ by $1$ .

$5 + (2 \cdot 2) - 2 \cdot - 2$

Step 13.2.5.36

Multiply $2$ by $2$ .

$5 + 4 - 2 \cdot - 2$

Step 13.2.5.37

Multiply $- 2$ by $- 2$ .

$5 + 4 + 4$

Step 13.2.5.38

Add $4$ and $4$ .

$5 + 8$

Step 13.2.5.39

Add $5$ and $8$ .

$13$

Step 14