Calculus Examples

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

Step 1

Remove parentheses.

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

Step 2

Since $- 3$ is constant with respect to $x$ , move $- 3$ out of the integral.

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

Step 3

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

$- 3 (\int_{- 3}^{0} - x d x + \int_{0}^{2} x d x)$

Step 4

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

$- 3 (- \int_{- 3}^{0} x d x + \int_{0}^{2} x d x)$

Step 5

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

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

Step 6

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

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

Step 7

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

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

Step 8

Simplify the answer.

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Step 8.1

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

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

Step 8.2

Substitute and simplify.

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Step 8.2.1

Evaluate $\frac{x^{2}}{2}$ at $0$ and at $- 3$ .

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

Step 8.2.2

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

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

Step 8.2.3

Simplify.

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Step 8.2.3.1

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

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

Step 8.2.3.2

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

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Step 8.2.3.2.1

Factor $2$ out of $0$ .

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

Step 8.2.3.2.2

Cancel the common factors.

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Step 8.2.3.2.2.1

Factor $2$ out of $2$ .

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

Step 8.2.3.2.2.2

Cancel the common factor.

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

Step 8.2.3.2.2.3

Rewrite the expression.

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

Step 8.2.3.2.2.4

Divide $0$ by $1$ .

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

Step 8.2.3.3

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

$- 3 (- (0 - \frac{9}{2}) + (\frac{2^{2}}{2}) - \frac{0^{2}}{2})$

Step 8.2.3.4

Subtract $\frac{9}{2}$ from $0$ .

$- 3 (- - \frac{9}{2} + (\frac{2^{2}}{2}) - \frac{0^{2}}{2})$

Step 8.2.3.5

Multiply $- 1$ by $- 1$ .

$- 3 (1 (\frac{9}{2}) + (\frac{2^{2}}{2}) - \frac{0^{2}}{2})$

Step 8.2.3.6

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

$- 3 (\frac{9}{2} + (\frac{2^{2}}{2}) - \frac{0^{2}}{2})$

Step 8.2.3.7

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

$- 3 (\frac{9}{2} + \frac{4}{2} - \frac{0^{2}}{2})$

Step 8.2.3.8

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

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Step 8.2.3.8.1

Factor $2$ out of $4$ .

$- 3 (\frac{9}{2} + \frac{2 \cdot 2}{2} - \frac{0^{2}}{2})$

Step 8.2.3.8.2

Cancel the common factors.

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Step 8.2.3.8.2.1

Factor $2$ out of $2$ .

$- 3 (\frac{9}{2} + \frac{2 \cdot 2}{2 (1)} - \frac{0^{2}}{2})$

Step 8.2.3.8.2.2

Cancel the common factor.

$- 3 (\frac{9}{2} + \frac{2 \cdot 2}{2 \cdot 1} - \frac{0^{2}}{2})$

Step 8.2.3.8.2.3

Rewrite the expression.

$- 3 (\frac{9}{2} + \frac{2}{1} - \frac{0^{2}}{2})$

Step 8.2.3.8.2.4

Divide $2$ by $1$ .

$- 3 (\frac{9}{2} + 2 - \frac{0^{2}}{2})$

Step 8.2.3.9

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

$- 3 (\frac{9}{2} + 2 - \frac{0}{2})$

Step 8.2.3.10

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

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Step 8.2.3.10.1

Factor $2$ out of $0$ .

$- 3 (\frac{9}{2} + 2 - \frac{2 (0)}{2})$

Step 8.2.3.10.2

Cancel the common factors.

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Step 8.2.3.10.2.1

Factor $2$ out of $2$ .

$- 3 (\frac{9}{2} + 2 - \frac{2 \cdot 0}{2 \cdot 1})$

Step 8.2.3.10.2.2

Cancel the common factor.

$- 3 (\frac{9}{2} + 2 - \frac{2 \cdot 0}{2 \cdot 1})$

Step 8.2.3.10.2.3

Rewrite the expression.

$- 3 (\frac{9}{2} + 2 - \frac{0}{1})$

Step 8.2.3.10.2.4

Divide $0$ by $1$ .

$- 3 (\frac{9}{2} + 2 - 0)$

Step 8.2.3.11

Multiply $- 1$ by $0$ .

$- 3 (\frac{9}{2} + 2 + 0)$

Step 8.2.3.12

Add $2$ and $0$ .

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

Step 8.2.3.13

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

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

Step 8.2.3.14

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

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

Step 8.2.3.15

Combine the numerators over the common denominator.

$- 3 \frac{9 + 2 \cdot 2}{2}$

Step 8.2.3.16

Simplify the numerator.

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Step 8.2.3.16.1

Multiply $2$ by $2$ .

$- 3 \frac{9 + 4}{2}$

Step 8.2.3.16.2

Add $9$ and $4$ .

$- 3 (\frac{13}{2})$

Step 8.2.3.17

Combine $- 3$ and $\frac{13}{2}$ .

$\frac{- 3 \cdot 13}{2}$

Step 8.2.3.18

Multiply $- 3$ by $13$ .

$\frac{- 39}{2}$

Step 8.2.3.19

Move the negative in front of the fraction.

$- \frac{39}{2}$

Step 9

The result can be shown in multiple forms.

Exact Form:

$- \frac{39}{2}$

Decimal Form:

$- 19.5$

Mixed Number Form:

$- 19 \frac{1}{2}$

Step 10