Calculus Examples

$\int_{1}^{4} (3 - | x - 3 |) d x$

Step 1

Remove parentheses.

$\int_{1}^{4} 3 - | x - 3 | d x$

Step 2

Split the single integral into multiple integrals.

$\int_{1}^{4} 3 d x + \int_{1}^{4} - | x - 3 | d x$

Step 3

Apply the constant rule.

$3 x]_{1}^{4} + \int_{1}^{4} - | x - 3 | d x$

Step 4

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

$3 x]_{1}^{4} - \int_{1}^{4} | x - 3 | d x$

Step 5

Split up the integral depending on where $x - 3$ is positive and negative.

$3 x]_{1}^{4} - (\int_{1}^{3} - x + 3 d x + \int_{3}^{4} x - 3 d x)$

Step 6

Split the single integral into multiple integrals.

$3 x]_{1}^{4} - (\int_{1}^{3} - x d x + \int_{1}^{3} 3 d x + \int_{3}^{4} x - 3 d x)$

Step 7

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

$3 x]_{1}^{4} - (- \int_{1}^{3} x d x + \int_{1}^{3} 3 d x + \int_{3}^{4} x - 3 d x)$

Step 8

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

$3 x]_{1}^{4} - (- (\frac{1}{2} x^{2}]_{1}^{3}) + \int_{1}^{3} 3 d x + \int_{3}^{4} x - 3 d x)$

Step 9

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

$3 x]_{1}^{4} - (- (\frac{x^{2}}{2}]_{1}^{3}) + \int_{1}^{3} 3 d x + \int_{3}^{4} x - 3 d x)$

Step 10

Apply the constant rule.

$3 x]_{1}^{4} - (- (\frac{x^{2}}{2}]_{1}^{3}) + 3 x]_{1}^{3} + \int_{3}^{4} x - 3 d x)$

Step 11

Split the single integral into multiple integrals.

$3 x]_{1}^{4} - (- (\frac{x^{2}}{2}]_{1}^{3}) + 3 x]_{1}^{3} + \int_{3}^{4} x d x + \int_{3}^{4} - 3 d x)$

Step 12

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

$3 x]_{1}^{4} - (- (\frac{x^{2}}{2}]_{1}^{3}) + 3 x]_{1}^{3} + \frac{1}{2} x^{2}]_{3}^{4} + \int_{3}^{4} - 3 d x)$

Step 13

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

$3 x]_{1}^{4} - (- (\frac{x^{2}}{2}]_{1}^{3}) + 3 x]_{1}^{3} + \frac{x^{2}}{2}]_{3}^{4} + \int_{3}^{4} - 3 d x)$

Step 14

Apply the constant rule.

$3 x]_{1}^{4} - (- (\frac{x^{2}}{2}]_{1}^{3}) + 3 x]_{1}^{3} + \frac{x^{2}}{2}]_{3}^{4} + - 3 x]_{3}^{4})$

Step 15

Simplify the answer.

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Step 15.1

Combine $\frac{x^{2}}{2}]_{3}^{4}$ and $- 3 x]_{3}^{4}$ .

$3 x]_{1}^{4} - (- (\frac{x^{2}}{2}]_{1}^{3}) + 3 x]_{1}^{3} + \frac{x^{2}}{2} - 3 x]_{3}^{4})$

Step 15.2

Substitute and simplify.

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Step 15.2.1

Evaluate $3 x$ at $4$ and at $1$ .

$(3 \cdot 4) - 3 \cdot 1 - (- (\frac{x^{2}}{2}]_{1}^{3}) + 3 x]_{1}^{3} + \frac{x^{2}}{2} - 3 x]_{3}^{4})$

Step 15.2.2

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

$(3 \cdot 4) - 3 \cdot 1 - (- ((\frac{3^{2}}{2}) - \frac{1^{2}}{2}) + 3 x]_{1}^{3} + \frac{x^{2}}{2} - 3 x]_{3}^{4})$

Step 15.2.3

Evaluate $3 x$ at $3$ and at $1$ .

$(3 \cdot 4) - 3 \cdot 1 - (- (\frac{3^{2}}{2} - \frac{1^{2}}{2}) + 3 \cdot 3 - 3 \cdot 1 + \frac{x^{2}}{2} - 3 x]_{3}^{4})$

Step 15.2.4

Evaluate $\frac{x^{2}}{2} - 3 x$ at $4$ and at $3$ .

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

Step 15.2.5

Simplify.

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Step 15.2.5.1

Multiply $3$ by $4$ .

$12 - 3 \cdot 1 - (- (\frac{3^{2}}{2} - \frac{1^{2}}{2}) + 3 \cdot 3 - 3 \cdot 1 + (\frac{4^{2}}{2} - 3 \cdot 4) - (\frac{3^{2}}{2} - 3 \cdot 3))$

Step 15.2.5.2

Multiply $- 3$ by $1$ .

$12 - 3 - (- (\frac{3^{2}}{2} - \frac{1^{2}}{2}) + 3 \cdot 3 - 3 \cdot 1 + (\frac{4^{2}}{2} - 3 \cdot 4) - (\frac{3^{2}}{2} - 3 \cdot 3))$

Step 15.2.5.3

Subtract $3$ from $12$ .

$9 - (- (\frac{3^{2}}{2} - \frac{1^{2}}{2}) + 3 \cdot 3 - 3 \cdot 1 + (\frac{4^{2}}{2} - 3 \cdot 4) - (\frac{3^{2}}{2} - 3 \cdot 3))$

Step 15.2.5.4

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

$9 - (- (\frac{9}{2} - \frac{1^{2}}{2}) + 3 \cdot 3 - 3 \cdot 1 + (\frac{4^{2}}{2} - 3 \cdot 4) - (\frac{3^{2}}{2} - 3 \cdot 3))$

Step 15.2.5.5

One to any power is one.

$9 - (- (\frac{9}{2} - \frac{1}{2}) + 3 \cdot 3 - 3 \cdot 1 + (\frac{4^{2}}{2} - 3 \cdot 4) - (\frac{3^{2}}{2} - 3 \cdot 3))$

Step 15.2.5.6

Combine the numerators over the common denominator.

$9 - (- \frac{9 - 1}{2} + 3 \cdot 3 - 3 \cdot 1 + (\frac{4^{2}}{2} - 3 \cdot 4) - (\frac{3^{2}}{2} - 3 \cdot 3))$

Step 15.2.5.7

Subtract $1$ from $9$ .

$9 - (- \frac{8}{2} + 3 \cdot 3 - 3 \cdot 1 + (\frac{4^{2}}{2} - 3 \cdot 4) - (\frac{3^{2}}{2} - 3 \cdot 3))$

Step 15.2.5.8

Cancel the common factor of $8$ and $2$ .

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Step 15.2.5.8.1

Factor $2$ out of $8$ .

$9 - (- \frac{2 \cdot 4}{2} + 3 \cdot 3 - 3 \cdot 1 + (\frac{4^{2}}{2} - 3 \cdot 4) - (\frac{3^{2}}{2} - 3 \cdot 3))$

Step 15.2.5.8.2

Cancel the common factors.

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Step 15.2.5.8.2.1

Factor $2$ out of $2$ .

$9 - (- \frac{2 \cdot 4}{2 (1)} + 3 \cdot 3 - 3 \cdot 1 + (\frac{4^{2}}{2} - 3 \cdot 4) - (\frac{3^{2}}{2} - 3 \cdot 3))$

Step 15.2.5.8.2.2

Cancel the common factor.

$9 - (- \frac{2 \cdot 4}{2 \cdot 1} + 3 \cdot 3 - 3 \cdot 1 + (\frac{4^{2}}{2} - 3 \cdot 4) - (\frac{3^{2}}{2} - 3 \cdot 3))$

Step 15.2.5.8.2.3

Rewrite the expression.

$9 - (- \frac{4}{1} + 3 \cdot 3 - 3 \cdot 1 + (\frac{4^{2}}{2} - 3 \cdot 4) - (\frac{3^{2}}{2} - 3 \cdot 3))$

Step 15.2.5.8.2.4

Divide $4$ by $1$ .

$9 - (- 1 \cdot 4 + 3 \cdot 3 - 3 \cdot 1 + (\frac{4^{2}}{2} - 3 \cdot 4) - (\frac{3^{2}}{2} - 3 \cdot 3))$

Step 15.2.5.9

Multiply $- 1$ by $4$ .

$9 - (- 4 + 3 \cdot 3 - 3 \cdot 1 + (\frac{4^{2}}{2} - 3 \cdot 4) - (\frac{3^{2}}{2} - 3 \cdot 3))$

Step 15.2.5.10

Multiply $3$ by $3$ .

$9 - (- 4 + 9 - 3 \cdot 1 + (\frac{4^{2}}{2} - 3 \cdot 4) - (\frac{3^{2}}{2} - 3 \cdot 3))$

Step 15.2.5.11

Multiply $- 3$ by $1$ .

$9 - (- 4 + 9 - 3 + (\frac{4^{2}}{2} - 3 \cdot 4) - (\frac{3^{2}}{2} - 3 \cdot 3))$

Step 15.2.5.12

Subtract $3$ from $9$ .

$9 - (- 4 + 6 + (\frac{4^{2}}{2} - 3 \cdot 4) - (\frac{3^{2}}{2} - 3 \cdot 3))$

Step 15.2.5.13

Add $- 4$ and $6$ .

$9 - (2 + (\frac{4^{2}}{2} - 3 \cdot 4) - (\frac{3^{2}}{2} - 3 \cdot 3))$

Step 15.2.5.14

Raise $4$ to the power of $2$ .

$9 - (2 + \frac{16}{2} - 3 \cdot 4 - (\frac{3^{2}}{2} - 3 \cdot 3))$

Step 15.2.5.15

Cancel the common factor of $16$ and $2$ .

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Step 15.2.5.15.1

Factor $2$ out of $16$ .

$9 - (2 + \frac{2 \cdot 8}{2} - 3 \cdot 4 - (\frac{3^{2}}{2} - 3 \cdot 3))$

Step 15.2.5.15.2

Cancel the common factors.

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Step 15.2.5.15.2.1

Factor $2$ out of $2$ .

$9 - (2 + \frac{2 \cdot 8}{2 (1)} - 3 \cdot 4 - (\frac{3^{2}}{2} - 3 \cdot 3))$

Step 15.2.5.15.2.2

Cancel the common factor.

$9 - (2 + \frac{2 \cdot 8}{2 \cdot 1} - 3 \cdot 4 - (\frac{3^{2}}{2} - 3 \cdot 3))$

Step 15.2.5.15.2.3

Rewrite the expression.

$9 - (2 + \frac{8}{1} - 3 \cdot 4 - (\frac{3^{2}}{2} - 3 \cdot 3))$

Step 15.2.5.15.2.4

Divide $8$ by $1$ .

$9 - (2 + 8 - 3 \cdot 4 - (\frac{3^{2}}{2} - 3 \cdot 3))$

Step 15.2.5.16

Multiply $- 3$ by $4$ .

$9 - (2 + 8 - 12 - (\frac{3^{2}}{2} - 3 \cdot 3))$

Step 15.2.5.17

Subtract $12$ from $8$ .

$9 - (2 - 4 - (\frac{3^{2}}{2} - 3 \cdot 3))$

Step 15.2.5.18

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

$9 - (2 - 4 - (\frac{9}{2} - 3 \cdot 3))$

Step 15.2.5.19

Multiply $- 3$ by $3$ .

$9 - (2 - 4 - (\frac{9}{2} - 9))$

Step 15.2.5.20

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

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

Step 15.2.5.21

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

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

Step 15.2.5.22

Combine the numerators over the common denominator.

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

Step 15.2.5.23

Simplify the numerator.

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Step 15.2.5.23.1

Multiply $- 9$ by $2$ .

$9 - (2 - 4 - \frac{9 - 18}{2})$

Step 15.2.5.23.2

Subtract $18$ from $9$ .

$9 - (2 - 4 - \frac{- 9}{2})$

Step 15.2.5.24

Move the negative in front of the fraction.

$9 - (2 - 4 - - \frac{9}{2})$

Step 15.2.5.25

Multiply $- 1$ by $- 1$ .

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

Step 15.2.5.26

Multiply $\frac{9}{2}$ by $1$ .

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

Step 15.2.5.27

To write $- 4$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

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

Step 15.2.5.28

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

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

Step 15.2.5.29

Combine the numerators over the common denominator.

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

Step 15.2.5.30

Simplify the numerator.

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Step 15.2.5.30.1

Multiply $- 4$ by $2$ .

$9 - (2 + \frac{- 8 + 9}{2})$

Step 15.2.5.30.2

Add $- 8$ and $9$ .

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

Step 15.2.5.31

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

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

Step 15.2.5.32

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

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

Step 15.2.5.33

Combine the numerators over the common denominator.

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

Step 15.2.5.34

Simplify the numerator.

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Step 15.2.5.34.1

Multiply $2$ by $2$ .

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

Step 15.2.5.34.2

Add $4$ and $1$ .

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

Step 15.2.5.35

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

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

Step 15.2.5.36

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

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

Step 15.2.5.37

Combine the numerators over the common denominator.

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

Step 15.2.5.38

Simplify the numerator.

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Step 15.2.5.38.1

Multiply $9$ by $2$ .

$\frac{18 - 5}{2}$

Step 15.2.5.38.2

Subtract $5$ from $18$ .

$\frac{13}{2}$

Step 16

The result can be shown in multiple forms.

Exact Form:

$\frac{13}{2}$

Decimal Form:

$6.5$

Mixed Number Form:

$6 \frac{1}{2}$

Step 17