Calculus Examples

$\int_{- 3}^{3} | x - 2 x - 3 | d x$

Step 1

Subtract $2 x$ from $x$ .

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

Step 2

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

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

Step 3

Split the single integral into multiple integrals.

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

Step 4

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

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

Step 5

Apply the constant rule.

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

Step 6

Simplify the answer.

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Step 6.1

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

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

Step 6.2

Substitute and simplify.

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Step 6.2.1

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

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

Step 6.2.2

Simplify.

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Step 6.2.2.1

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

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

Step 6.2.2.2

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

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

Step 6.2.2.3

Multiply $3$ by $3$ .

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

Step 6.2.2.4

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

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

Step 6.2.2.5

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

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

Step 6.2.2.6

Combine the numerators over the common denominator.

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

Step 6.2.2.7

Simplify the numerator.

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Step 6.2.2.7.1

Multiply $9$ by $2$ .

$\frac{9 + 18}{2} - (\frac{1}{2} {(- 3)}^{2} + 3 \cdot - 3)$

Step 6.2.2.7.2

Add $9$ and $18$ .

$\frac{27}{2} - (\frac{1}{2} {(- 3)}^{2} + 3 \cdot - 3)$

Step 6.2.2.8

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

$\frac{27}{2} - (\frac{1}{2} \cdot 9 + 3 \cdot - 3)$

Step 6.2.2.9

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

$\frac{27}{2} - (\frac{9}{2} + 3 \cdot - 3)$

Step 6.2.2.10

Multiply $3$ by $- 3$ .

$\frac{27}{2} - (\frac{9}{2} - 9)$

Step 6.2.2.11

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

$\frac{27}{2} - (\frac{9}{2} - 9 \cdot \frac{2}{2})$

Step 6.2.2.12

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

$\frac{27}{2} - (\frac{9}{2} + \frac{- 9 \cdot 2}{2})$

Step 6.2.2.13

Combine the numerators over the common denominator.

$\frac{27}{2} - \frac{9 - 9 \cdot 2}{2}$

Step 6.2.2.14

Simplify the numerator.

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Step 6.2.2.14.1

Multiply $- 9$ by $2$ .

$\frac{27}{2} - \frac{9 - 18}{2}$

Step 6.2.2.14.2

Subtract $18$ from $9$ .

$\frac{27}{2} - \frac{- 9}{2}$

Step 6.2.2.15

Move the negative in front of the fraction.

$\frac{27}{2} - - \frac{9}{2}$

Step 6.2.2.16

Multiply $- 1$ by $- 1$ .

$\frac{27}{2} + 1 (\frac{9}{2})$

Step 6.2.2.17

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

$\frac{27}{2} + \frac{9}{2}$

Step 6.2.2.18

Combine the numerators over the common denominator.

$\frac{27 + 9}{2}$

Step 6.2.2.19

Add $27$ and $9$ .

$\frac{36}{2}$

Step 6.2.2.20

Cancel the common factor of $36$ and $2$ .

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Step 6.2.2.20.1

Factor $2$ out of $36$ .

$\frac{2 \cdot 18}{2}$

Step 6.2.2.20.2

Cancel the common factors.

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Step 6.2.2.20.2.1

Factor $2$ out of $2$ .

$\frac{2 \cdot 18}{2 (1)}$

Step 6.2.2.20.2.2

Cancel the common factor.

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

Step 6.2.2.20.2.3

Rewrite the expression.

$\frac{18}{1}$

Step 6.2.2.20.2.4

Divide $18$ by $1$ .

$18$

Step 7