Calculus Examples

$\int_{- 1}^{2} (x - 8 | x |) d x$

Step 1

Remove parentheses.

$\int_{- 1}^{2} x - 8 | x | d x$

Step 2

Split the single integral into multiple integrals.

$\int_{- 1}^{2} x d x + \int_{- 1}^{2} - 8 | x | d x$

Step 3

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

$\frac{1}{2} x^{2}]_{- 1}^{2} + \int_{- 1}^{2} - 8 | x | d x$

Step 4

Since $- 8$ is constant with respect to $x$ , move $- 8$ out of the integral.

$\frac{1}{2} x^{2}]_{- 1}^{2} - 8 \int_{- 1}^{2} | x | d x$

Step 5

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

$\frac{1}{2} x^{2}]_{- 1}^{2} - 8 (\int_{- 1}^{0} - x d x + \int_{0}^{2} x d x)$

Step 6

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

$\frac{1}{2} x^{2}]_{- 1}^{2} - 8 (- \int_{- 1}^{0} x d x + \int_{0}^{2} x d x)$

Step 7

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

$\frac{1}{2} x^{2}]_{- 1}^{2} - 8 (- (\frac{1}{2} x^{2}]_{- 1}^{0}) + \int_{0}^{2} x d x)$

Step 8

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

$\frac{1}{2} x^{2}]_{- 1}^{2} - 8 (- (\frac{x^{2}}{2}]_{- 1}^{0}) + \int_{0}^{2} x d x)$

Step 9

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

$\frac{1}{2} x^{2}]_{- 1}^{2} - 8 (- (\frac{x^{2}}{2}]_{- 1}^{0}) + \frac{1}{2} x^{2}]_{0}^{2})$

Step 10

Simplify the answer.

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Step 10.1

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

$\frac{1}{2} x^{2}]_{- 1}^{2} - 8 (- (\frac{x^{2}}{2}]_{- 1}^{0}) + \frac{x^{2}}{2}]_{0}^{2})$

Step 10.2

Substitute and simplify.

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Step 10.2.1

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

$(\frac{1}{2} \cdot 2^{2}) - \frac{1}{2} {(- 1)}^{2} - 8 (- (\frac{x^{2}}{2}]_{- 1}^{0}) + \frac{x^{2}}{2}]_{0}^{2})$

Step 10.2.2

Evaluate $\frac{x^{2}}{2}$ at $0$ and at $- 1$ .

$(\frac{1}{2} \cdot 2^{2}) - \frac{1}{2} {(- 1)}^{2} - 8 (- ((\frac{0^{2}}{2}) - \frac{{(- 1)}^{2}}{2}) + \frac{x^{2}}{2}]_{0}^{2})$

Step 10.2.3

Evaluate $\frac{x^{2}}{2}$ at $2$ and at $0$ .

$(\frac{1}{2} \cdot 2^{2}) - \frac{1}{2} {(- 1)}^{2} - 8 (- ((\frac{0^{2}}{2}) - \frac{{(- 1)}^{2}}{2}) + (\frac{2^{2}}{2}) - \frac{0^{2}}{2})$

Step 10.2.4

Simplify.

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Step 10.2.4.1

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

$\frac{1}{2} \cdot 4 - \frac{1}{2} {(- 1)}^{2} - 8 (- ((\frac{0^{2}}{2}) - \frac{{(- 1)}^{2}}{2}) + (\frac{2^{2}}{2}) - \frac{0^{2}}{2})$

Step 10.2.4.2

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

$\frac{4}{2} - \frac{1}{2} {(- 1)}^{2} - 8 (- ((\frac{0^{2}}{2}) - \frac{{(- 1)}^{2}}{2}) + (\frac{2^{2}}{2}) - \frac{0^{2}}{2})$

Step 10.2.4.3

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

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Step 10.2.4.3.1

Factor $2$ out of $4$ .

$\frac{2 \cdot 2}{2} - \frac{1}{2} {(- 1)}^{2} - 8 (- ((\frac{0^{2}}{2}) - \frac{{(- 1)}^{2}}{2}) + (\frac{2^{2}}{2}) - \frac{0^{2}}{2})$

Step 10.2.4.3.2

Cancel the common factors.

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Step 10.2.4.3.2.1

Factor $2$ out of $2$ .

$\frac{2 \cdot 2}{2 (1)} - \frac{1}{2} {(- 1)}^{2} - 8 (- ((\frac{0^{2}}{2}) - \frac{{(- 1)}^{2}}{2}) + (\frac{2^{2}}{2}) - \frac{0^{2}}{2})$

Step 10.2.4.3.2.2

Cancel the common factor.

$\frac{2 \cdot 2}{2 \cdot 1} - \frac{1}{2} {(- 1)}^{2} - 8 (- ((\frac{0^{2}}{2}) - \frac{{(- 1)}^{2}}{2}) + (\frac{2^{2}}{2}) - \frac{0^{2}}{2})$

Step 10.2.4.3.2.3

Rewrite the expression.

$\frac{2}{1} - \frac{1}{2} {(- 1)}^{2} - 8 (- ((\frac{0^{2}}{2}) - \frac{{(- 1)}^{2}}{2}) + (\frac{2^{2}}{2}) - \frac{0^{2}}{2})$

Step 10.2.4.3.2.4

Divide $2$ by $1$ .

$2 - \frac{1}{2} {(- 1)}^{2} - 8 (- ((\frac{0^{2}}{2}) - \frac{{(- 1)}^{2}}{2}) + (\frac{2^{2}}{2}) - \frac{0^{2}}{2})$

Step 10.2.4.4

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

$2 - \frac{1}{2} \cdot 1 - 8 (- ((\frac{0^{2}}{2}) - \frac{{(- 1)}^{2}}{2}) + (\frac{2^{2}}{2}) - \frac{0^{2}}{2})$

Step 10.2.4.5

Multiply $- 1$ by $1$ .

$2 - \frac{1}{2} - 8 (- ((\frac{0^{2}}{2}) - \frac{{(- 1)}^{2}}{2}) + (\frac{2^{2}}{2}) - \frac{0^{2}}{2})$

Step 10.2.4.6

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

$2 \cdot \frac{2}{2} - \frac{1}{2} - 8 (- ((\frac{0^{2}}{2}) - \frac{{(- 1)}^{2}}{2}) + (\frac{2^{2}}{2}) - \frac{0^{2}}{2})$

Step 10.2.4.7

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

$\frac{2 \cdot 2}{2} - \frac{1}{2} - 8 (- ((\frac{0^{2}}{2}) - \frac{{(- 1)}^{2}}{2}) + (\frac{2^{2}}{2}) - \frac{0^{2}}{2})$

Step 10.2.4.8

Combine the numerators over the common denominator.

$\frac{2 \cdot 2 - 1}{2} - 8 (- ((\frac{0^{2}}{2}) - \frac{{(- 1)}^{2}}{2}) + (\frac{2^{2}}{2}) - \frac{0^{2}}{2})$

Step 10.2.4.9

Simplify the numerator.

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Step 10.2.4.9.1

Multiply $2$ by $2$ .

$\frac{4 - 1}{2} - 8 (- ((\frac{0^{2}}{2}) - \frac{{(- 1)}^{2}}{2}) + (\frac{2^{2}}{2}) - \frac{0^{2}}{2})$

Step 10.2.4.9.2

Subtract $1$ from $4$ .

$\frac{3}{2} - 8 (- ((\frac{0^{2}}{2}) - \frac{{(- 1)}^{2}}{2}) + (\frac{2^{2}}{2}) - \frac{0^{2}}{2})$

Step 10.2.4.10

Raising $0$ to any positive power yields $0$ .

$\frac{3}{2} - 8 (- (\frac{0}{2} - \frac{{(- 1)}^{2}}{2}) + (\frac{2^{2}}{2}) - \frac{0^{2}}{2})$

Step 10.2.4.11

Cancel the common factor of $0$ and $2$ .

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Step 10.2.4.11.1

Factor $2$ out of $0$ .

$\frac{3}{2} - 8 (- (\frac{2 (0)}{2} - \frac{{(- 1)}^{2}}{2}) + (\frac{2^{2}}{2}) - \frac{0^{2}}{2})$

Step 10.2.4.11.2

Cancel the common factors.

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Step 10.2.4.11.2.1

Factor $2$ out of $2$ .

$\frac{3}{2} - 8 (- (\frac{2 \cdot 0}{2 \cdot 1} - \frac{{(- 1)}^{2}}{2}) + (\frac{2^{2}}{2}) - \frac{0^{2}}{2})$

Step 10.2.4.11.2.2

Cancel the common factor.

$\frac{3}{2} - 8 (- (\frac{2 \cdot 0}{2 \cdot 1} - \frac{{(- 1)}^{2}}{2}) + (\frac{2^{2}}{2}) - \frac{0^{2}}{2})$

Step 10.2.4.11.2.3

Rewrite the expression.

$\frac{3}{2} - 8 (- (\frac{0}{1} - \frac{{(- 1)}^{2}}{2}) + (\frac{2^{2}}{2}) - \frac{0^{2}}{2})$

Step 10.2.4.11.2.4

Divide $0$ by $1$ .

$\frac{3}{2} - 8 (- (0 - \frac{{(- 1)}^{2}}{2}) + (\frac{2^{2}}{2}) - \frac{0^{2}}{2})$

Step 10.2.4.12

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

$\frac{3}{2} - 8 (- (0 - \frac{1}{2}) + (\frac{2^{2}}{2}) - \frac{0^{2}}{2})$

Step 10.2.4.13

Subtract $\frac{1}{2}$ from $0$ .

$\frac{3}{2} - 8 (- - \frac{1}{2} + (\frac{2^{2}}{2}) - \frac{0^{2}}{2})$

Step 10.2.4.14

Multiply $- 1$ by $- 1$ .

$\frac{3}{2} - 8 (1 (\frac{1}{2}) + (\frac{2^{2}}{2}) - \frac{0^{2}}{2})$

Step 10.2.4.15

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

$\frac{3}{2} - 8 (\frac{1}{2} + (\frac{2^{2}}{2}) - \frac{0^{2}}{2})$

Step 10.2.4.16

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

$\frac{3}{2} - 8 (\frac{1}{2} + \frac{4}{2} - \frac{0^{2}}{2})$

Step 10.2.4.17

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

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Step 10.2.4.17.1

Factor $2$ out of $4$ .

$\frac{3}{2} - 8 (\frac{1}{2} + \frac{2 \cdot 2}{2} - \frac{0^{2}}{2})$

Step 10.2.4.17.2

Cancel the common factors.

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Step 10.2.4.17.2.1

Factor $2$ out of $2$ .

$\frac{3}{2} - 8 (\frac{1}{2} + \frac{2 \cdot 2}{2 (1)} - \frac{0^{2}}{2})$

Step 10.2.4.17.2.2

Cancel the common factor.

$\frac{3}{2} - 8 (\frac{1}{2} + \frac{2 \cdot 2}{2 \cdot 1} - \frac{0^{2}}{2})$

Step 10.2.4.17.2.3

Rewrite the expression.

$\frac{3}{2} - 8 (\frac{1}{2} + \frac{2}{1} - \frac{0^{2}}{2})$

Step 10.2.4.17.2.4

Divide $2$ by $1$ .

$\frac{3}{2} - 8 (\frac{1}{2} + 2 - \frac{0^{2}}{2})$

Step 10.2.4.18

Raising $0$ to any positive power yields $0$ .

$\frac{3}{2} - 8 (\frac{1}{2} + 2 - \frac{0}{2})$

Step 10.2.4.19

Cancel the common factor of $0$ and $2$ .

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Step 10.2.4.19.1

Factor $2$ out of $0$ .

$\frac{3}{2} - 8 (\frac{1}{2} + 2 - \frac{2 (0)}{2})$

Step 10.2.4.19.2

Cancel the common factors.

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Step 10.2.4.19.2.1

Factor $2$ out of $2$ .

$\frac{3}{2} - 8 (\frac{1}{2} + 2 - \frac{2 \cdot 0}{2 \cdot 1})$

Step 10.2.4.19.2.2

Cancel the common factor.

$\frac{3}{2} - 8 (\frac{1}{2} + 2 - \frac{2 \cdot 0}{2 \cdot 1})$

Step 10.2.4.19.2.3

Rewrite the expression.

$\frac{3}{2} - 8 (\frac{1}{2} + 2 - \frac{0}{1})$

Step 10.2.4.19.2.4

Divide $0$ by $1$ .

$\frac{3}{2} - 8 (\frac{1}{2} + 2 - 0)$

Step 10.2.4.20

Multiply $- 1$ by $0$ .

$\frac{3}{2} - 8 (\frac{1}{2} + 2 + 0)$

Step 10.2.4.21

Add $2$ and $0$ .

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

Step 10.2.4.22

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

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

Step 10.2.4.23

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

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

Step 10.2.4.24

Combine the numerators over the common denominator.

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

Step 10.2.4.25

Simplify the numerator.

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Step 10.2.4.25.1

Multiply $2$ by $2$ .

$\frac{3}{2} - 8 \frac{1 + 4}{2}$

Step 10.2.4.25.2

Add $1$ and $4$ .

$\frac{3}{2} - 8 (\frac{5}{2})$

Step 10.2.4.26

Combine $- 8$ and $\frac{5}{2}$ .

$\frac{3}{2} + \frac{- 8 \cdot 5}{2}$

Step 10.2.4.27

Multiply $- 8$ by $5$ .

$\frac{3}{2} + \frac{- 40}{2}$

Step 10.2.4.28

Cancel the common factor of $- 40$ and $2$ .

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Step 10.2.4.28.1

Factor $2$ out of $- 40$ .

$\frac{3}{2} + \frac{2 \cdot - 20}{2}$

Step 10.2.4.28.2

Cancel the common factors.

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Step 10.2.4.28.2.1

Factor $2$ out of $2$ .

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

Step 10.2.4.28.2.2

Cancel the common factor.

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

Step 10.2.4.28.2.3

Rewrite the expression.

$\frac{3}{2} + \frac{- 20}{1}$

Step 10.2.4.28.2.4

Divide $- 20$ by $1$ .

$\frac{3}{2} - 20$

Step 10.2.4.29

To write $- 20$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$\frac{3}{2} - 20 \cdot \frac{2}{2}$

Step 10.2.4.30

Combine $- 20$ and $\frac{2}{2}$ .

$\frac{3}{2} + \frac{- 20 \cdot 2}{2}$

Step 10.2.4.31

Combine the numerators over the common denominator.

$\frac{3 - 20 \cdot 2}{2}$

Step 10.2.4.32

Simplify the numerator.

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Step 10.2.4.32.1

Multiply $- 20$ by $2$ .

$\frac{3 - 40}{2}$

Step 10.2.4.32.2

Subtract $40$ from $3$ .

$\frac{- 37}{2}$

Step 10.2.4.33

Move the negative in front of the fraction.

$- \frac{37}{2}$

Step 11

The result can be shown in multiple forms.

Exact Form:

$- \frac{37}{2}$

Decimal Form:

$- 18.5$

Mixed Number Form:

$- 18 \frac{1}{2}$

Step 12