Calculus Examples

$\int_{0}^{3} | x - 1 | d x$

Step 1

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

$\int_{0}^{1} - x + 1 d x + \int_{1}^{3} x - 1 d x$

Step 2

Split the single integral into multiple integrals.

$\int_{0}^{1} - x d x + \int_{0}^{1} d x + \int_{1}^{3} x - 1 d x$

Step 3

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

$- \int_{0}^{1} x d x + \int_{0}^{1} d x + \int_{1}^{3} x - 1 d x$

Step 4

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

$- (\frac{1}{2} x^{2}]_{0}^{1}) + \int_{0}^{1} d x + \int_{1}^{3} x - 1 d x$

Step 5

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

$- (\frac{x^{2}}{2}]_{0}^{1}) + \int_{0}^{1} d x + \int_{1}^{3} x - 1 d x$

Step 6

Apply the constant rule.

$- (\frac{x^{2}}{2}]_{0}^{1}) + x]_{0}^{1} + \int_{1}^{3} x - 1 d x$

Step 7

Split the single integral into multiple integrals.

$- (\frac{x^{2}}{2}]_{0}^{1}) + x]_{0}^{1} + \int_{1}^{3} x d x + \int_{1}^{3} - 1 d x$

Step 8

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

$- (\frac{x^{2}}{2}]_{0}^{1}) + x]_{0}^{1} + \frac{1}{2} x^{2}]_{1}^{3} + \int_{1}^{3} - 1 d x$

Step 9

Apply the constant rule.

$- (\frac{x^{2}}{2}]_{0}^{1}) + x]_{0}^{1} + \frac{1}{2} x^{2}]_{1}^{3} + - x]_{1}^{3}$

Step 10

Simplify the answer.

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Step 10.1

Combine $\frac{1}{2} x^{2}]_{1}^{3}$ and $- x]_{1}^{3}$ .

$- (\frac{x^{2}}{2}]_{0}^{1}) + x]_{0}^{1} + \frac{1}{2} x^{2} - x]_{1}^{3}$

Step 10.2

Substitute and simplify.

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Step 10.2.1

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

$- ((\frac{1^{2}}{2}) - \frac{0^{2}}{2}) + x]_{0}^{1} + \frac{1}{2} x^{2} - x]_{1}^{3}$

Step 10.2.2

Evaluate $x$ at $1$ and at $0$ .

$- (\frac{1^{2}}{2} - \frac{0^{2}}{2}) + 1 + 0 + \frac{1}{2} x^{2} - x]_{1}^{3}$

Step 10.2.3

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

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

Step 10.2.4

Simplify.

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Step 10.2.4.1

One to any power is one.

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

Step 10.2.4.2

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

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

Step 10.2.4.3

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

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Step 10.2.4.3.1

Factor $2$ out of $0$ .

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

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 $2$ .

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

Step 10.2.4.3.2.2

Cancel the common factor.

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

Step 10.2.4.3.2.3

Rewrite the expression.

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

Step 10.2.4.3.2.4

Divide $0$ by $1$ .

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

Step 10.2.4.4

Multiply $- 1$ by $0$ .

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

Step 10.2.4.5

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

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

Step 10.2.4.6

Add $1$ and $0$ .

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

Step 10.2.4.7

Write $1$ as a fraction with a common denominator.

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

Step 10.2.4.8

Combine the numerators over the common denominator.

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

Step 10.2.4.9

Add $- 1$ and $2$ .

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

Step 10.2.4.10

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

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

Step 10.2.4.11

Combine $\frac{1}{2}$ and $9$ .

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

Step 10.2.4.12

Multiply $- 1$ by $3$ .

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

Step 10.2.4.13

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

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

Step 10.2.4.14

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

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

Step 10.2.4.15

Combine the numerators over the common denominator.

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

Step 10.2.4.16

Simplify the numerator.

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Step 10.2.4.16.1

Multiply $- 3$ by $2$ .

$\frac{1}{2} + \frac{9 - 6}{2} - (\frac{1}{2} \cdot 1^{2} - 1 \cdot 1)$

Step 10.2.4.16.2

Subtract $6$ from $9$ .

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

Step 10.2.4.17

One to any power is one.

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

Step 10.2.4.18

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

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

Step 10.2.4.19

Multiply $- 1$ by $1$ .

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

Step 10.2.4.20

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

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

Step 10.2.4.21

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

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

Step 10.2.4.22

Combine the numerators over the common denominator.

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

Step 10.2.4.23

Simplify the numerator.

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Step 10.2.4.23.1

Multiply $- 1$ by $2$ .

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

Step 10.2.4.23.2

Subtract $2$ from $1$ .

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

Step 10.2.4.24

Move the negative in front of the fraction.

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

Step 10.2.4.25

Multiply $- 1$ by $- 1$ .

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

Step 10.2.4.26

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

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

Step 10.2.4.27

Combine the numerators over the common denominator.

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

Step 10.2.4.28

Add $3$ and $1$ .

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

Step 10.2.4.29

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

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Step 10.2.4.29.1

Factor $2$ out of $4$ .

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

Step 10.2.4.29.2

Cancel the common factors.

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Step 10.2.4.29.2.1

Factor $2$ out of $2$ .

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

Step 10.2.4.29.2.2

Cancel the common factor.

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

Step 10.2.4.29.2.3

Rewrite the expression.

$\frac{1}{2} + \frac{2}{1}$

Step 10.2.4.29.2.4

Divide $2$ by $1$ .

$\frac{1}{2} + 2$

Step 10.2.4.30

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

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

Step 10.2.4.31

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

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

Step 10.2.4.32

Combine the numerators over the common denominator.

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

Step 10.2.4.33

Simplify the numerator.

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Step 10.2.4.33.1

Multiply $2$ by $2$ .

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

Step 10.2.4.33.2

Add $1$ and $4$ .

$\frac{5}{2}$

Step 11

The result can be shown in multiple forms.

Exact Form:

$\frac{5}{2}$

Decimal Form:

$2.5$

Mixed Number Form:

$2 \frac{1}{2}$

Step 12