Calculus Examples

$\int_{- 3}^{2} - x^{2} - x + 6 d x$

Step 1

Split the single integral into multiple integrals.

$\int_{- 3}^{2} - x^{2} d x + \int_{- 3}^{2} - x d x + \int_{- 3}^{2} 6 d x$

Step 2

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

$- \int_{- 3}^{2} x^{2} d x + \int_{- 3}^{2} - x d x + \int_{- 3}^{2} 6 d x$

Step 3

By the Power Rule, the integral of $x^{2}$ with respect to $x$ is $\frac{1}{3} x^{3}$ .

$- (\frac{1}{3} x^{3}]_{- 3}^{2}) + \int_{- 3}^{2} - x d x + \int_{- 3}^{2} 6 d x$

Step 4

Combine $\frac{1}{3}$ and $x^{3}$ .

$- (\frac{x^{3}}{3}]_{- 3}^{2}) + \int_{- 3}^{2} - x d x + \int_{- 3}^{2} 6 d x$

Step 5

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

$- (\frac{x^{3}}{3}]_{- 3}^{2}) - \int_{- 3}^{2} x d x + \int_{- 3}^{2} 6 d x$

Step 6

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

$- (\frac{x^{3}}{3}]_{- 3}^{2}) - (\frac{1}{2} x^{2}]_{- 3}^{2}) + \int_{- 3}^{2} 6 d x$

Step 7

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

$- (\frac{x^{3}}{3}]_{- 3}^{2}) - (\frac{x^{2}}{2}]_{- 3}^{2}) + \int_{- 3}^{2} 6 d x$

Step 8

Apply the constant rule.

$- (\frac{x^{3}}{3}]_{- 3}^{2}) - (\frac{x^{2}}{2}]_{- 3}^{2}) + 6 x]_{- 3}^{2}$

Step 9

Substitute and simplify.

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Step 9.1

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

$- ((\frac{2^{3}}{3}) - \frac{{(- 3)}^{3}}{3}) - (\frac{x^{2}}{2}]_{- 3}^{2}) + 6 x]_{- 3}^{2}$

Step 9.2

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

$- (\frac{2^{3}}{3} - \frac{{(- 3)}^{3}}{3}) - (\frac{2^{2}}{2} - \frac{{(- 3)}^{2}}{2}) + 6 x]_{- 3}^{2}$

Step 9.3

Evaluate $6 x$ at $2$ and at $- 3$ .

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

Step 9.4

Simplify.

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Step 9.4.1

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

$- (\frac{8}{3} - \frac{{(- 3)}^{3}}{3}) - (\frac{2^{2}}{2} - \frac{{(- 3)}^{2}}{2}) + (6 \cdot 2) - 6 \cdot - 3$

Step 9.4.2

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

$- (\frac{8}{3} - \frac{- 27}{3}) - (\frac{2^{2}}{2} - \frac{{(- 3)}^{2}}{2}) + (6 \cdot 2) - 6 \cdot - 3$

Step 9.4.3

Cancel the common factor of $- 27$ and $3$ .

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Step 9.4.3.1

Factor $3$ out of $- 27$ .

$- (\frac{8}{3} - \frac{3 \cdot - 9}{3}) - (\frac{2^{2}}{2} - \frac{{(- 3)}^{2}}{2}) + (6 \cdot 2) - 6 \cdot - 3$

Step 9.4.3.2

Cancel the common factors.

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Step 9.4.3.2.1

Factor $3$ out of $3$ .

$- (\frac{8}{3} - \frac{3 \cdot - 9}{3 (1)}) - (\frac{2^{2}}{2} - \frac{{(- 3)}^{2}}{2}) + (6 \cdot 2) - 6 \cdot - 3$

Step 9.4.3.2.2

Cancel the common factor.

$- (\frac{8}{3} - \frac{3 \cdot - 9}{3 \cdot 1}) - (\frac{2^{2}}{2} - \frac{{(- 3)}^{2}}{2}) + (6 \cdot 2) - 6 \cdot - 3$

Step 9.4.3.2.3

Rewrite the expression.

$- (\frac{8}{3} - \frac{- 9}{1}) - (\frac{2^{2}}{2} - \frac{{(- 3)}^{2}}{2}) + (6 \cdot 2) - 6 \cdot - 3$

Step 9.4.3.2.4

Divide $- 9$ by $1$ .

$- (\frac{8}{3} - - 9) - (\frac{2^{2}}{2} - \frac{{(- 3)}^{2}}{2}) + (6 \cdot 2) - 6 \cdot - 3$

Step 9.4.4

Multiply $- 1$ by $- 9$ .

$- (\frac{8}{3} + 9) - (\frac{2^{2}}{2} - \frac{{(- 3)}^{2}}{2}) + (6 \cdot 2) - 6 \cdot - 3$

Step 9.4.5

To write $9$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$- (\frac{8}{3} + 9 \cdot \frac{3}{3}) - (\frac{2^{2}}{2} - \frac{{(- 3)}^{2}}{2}) + (6 \cdot 2) - 6 \cdot - 3$

Step 9.4.6

Combine $9$ and $\frac{3}{3}$ .

$- (\frac{8}{3} + \frac{9 \cdot 3}{3}) - (\frac{2^{2}}{2} - \frac{{(- 3)}^{2}}{2}) + (6 \cdot 2) - 6 \cdot - 3$

Step 9.4.7

Combine the numerators over the common denominator.

$- \frac{8 + 9 \cdot 3}{3} - (\frac{2^{2}}{2} - \frac{{(- 3)}^{2}}{2}) + (6 \cdot 2) - 6 \cdot - 3$

Step 9.4.8

Simplify the numerator.

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Step 9.4.8.1

Multiply $9$ by $3$ .

$- \frac{8 + 27}{3} - (\frac{2^{2}}{2} - \frac{{(- 3)}^{2}}{2}) + (6 \cdot 2) - 6 \cdot - 3$

Step 9.4.8.2

Add $8$ and $27$ .

$- \frac{35}{3} - (\frac{2^{2}}{2} - \frac{{(- 3)}^{2}}{2}) + (6 \cdot 2) - 6 \cdot - 3$

Step 9.4.9

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

$- \frac{35}{3} - (\frac{4}{2} - \frac{{(- 3)}^{2}}{2}) + (6 \cdot 2) - 6 \cdot - 3$

Step 9.4.10

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

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Step 9.4.10.1

Factor $2$ out of $4$ .

$- \frac{35}{3} - (\frac{2 \cdot 2}{2} - \frac{{(- 3)}^{2}}{2}) + (6 \cdot 2) - 6 \cdot - 3$

Step 9.4.10.2

Cancel the common factors.

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Step 9.4.10.2.1

Factor $2$ out of $2$ .

$- \frac{35}{3} - (\frac{2 \cdot 2}{2 (1)} - \frac{{(- 3)}^{2}}{2}) + (6 \cdot 2) - 6 \cdot - 3$

Step 9.4.10.2.2

Cancel the common factor.

$- \frac{35}{3} - (\frac{2 \cdot 2}{2 \cdot 1} - \frac{{(- 3)}^{2}}{2}) + (6 \cdot 2) - 6 \cdot - 3$

Step 9.4.10.2.3

Rewrite the expression.

$- \frac{35}{3} - (\frac{2}{1} - \frac{{(- 3)}^{2}}{2}) + (6 \cdot 2) - 6 \cdot - 3$

Step 9.4.10.2.4

Divide $2$ by $1$ .

$- \frac{35}{3} - (2 - \frac{{(- 3)}^{2}}{2}) + (6 \cdot 2) - 6 \cdot - 3$

Step 9.4.11

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

$- \frac{35}{3} - (2 - \frac{9}{2}) + (6 \cdot 2) - 6 \cdot - 3$

Step 9.4.12

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

$- \frac{35}{3} - (2 \cdot \frac{2}{2} - \frac{9}{2}) + (6 \cdot 2) - 6 \cdot - 3$

Step 9.4.13

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

$- \frac{35}{3} - (\frac{2 \cdot 2}{2} - \frac{9}{2}) + (6 \cdot 2) - 6 \cdot - 3$

Step 9.4.14

Combine the numerators over the common denominator.

$- \frac{35}{3} - \frac{2 \cdot 2 - 9}{2} + (6 \cdot 2) - 6 \cdot - 3$

Step 9.4.15

Simplify the numerator.

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Step 9.4.15.1

Multiply $2$ by $2$ .

$- \frac{35}{3} - \frac{4 - 9}{2} + (6 \cdot 2) - 6 \cdot - 3$

Step 9.4.15.2

Subtract $9$ from $4$ .

$- \frac{35}{3} - \frac{- 5}{2} + (6 \cdot 2) - 6 \cdot - 3$

Step 9.4.16

Move the negative in front of the fraction.

$- \frac{35}{3} - - \frac{5}{2} + (6 \cdot 2) - 6 \cdot - 3$

Step 9.4.17

Multiply $- 1$ by $- 1$ .

$- \frac{35}{3} + 1 (\frac{5}{2}) + (6 \cdot 2) - 6 \cdot - 3$

Step 9.4.18

Multiply $\frac{5}{2}$ by $1$ .

$- \frac{35}{3} + \frac{5}{2} + (6 \cdot 2) - 6 \cdot - 3$

Step 9.4.19

To write $- \frac{35}{3}$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$- \frac{35}{3} \cdot \frac{2}{2} + \frac{5}{2} + (6 \cdot 2) - 6 \cdot - 3$

Step 9.4.20

To write $\frac{5}{2}$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$- \frac{35}{3} \cdot \frac{2}{2} + \frac{5}{2} \cdot \frac{3}{3} + (6 \cdot 2) - 6 \cdot - 3$

Step 9.4.21

Write each expression with a common denominator of $6$ , by multiplying each by an appropriate factor of $1$ .

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Step 9.4.21.1

Multiply $\frac{35}{3}$ by $\frac{2}{2}$ .

$- \frac{35 \cdot 2}{3 \cdot 2} + \frac{5}{2} \cdot \frac{3}{3} + (6 \cdot 2) - 6 \cdot - 3$

Step 9.4.21.2

Multiply $3$ by $2$ .

$- \frac{35 \cdot 2}{6} + \frac{5}{2} \cdot \frac{3}{3} + (6 \cdot 2) - 6 \cdot - 3$

Step 9.4.21.3

Multiply $\frac{5}{2}$ by $\frac{3}{3}$ .

$- \frac{35 \cdot 2}{6} + \frac{5 \cdot 3}{2 \cdot 3} + (6 \cdot 2) - 6 \cdot - 3$

Step 9.4.21.4

Multiply $2$ by $3$ .

$- \frac{35 \cdot 2}{6} + \frac{5 \cdot 3}{6} + (6 \cdot 2) - 6 \cdot - 3$

Step 9.4.22

Combine the numerators over the common denominator.

$\frac{- 35 \cdot 2 + 5 \cdot 3}{6} + (6 \cdot 2) - 6 \cdot - 3$

Step 9.4.23

Simplify the numerator.

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Step 9.4.23.1

Multiply $- 35$ by $2$ .

$\frac{- 70 + 5 \cdot 3}{6} + (6 \cdot 2) - 6 \cdot - 3$

Step 9.4.23.2

Multiply $5$ by $3$ .

$\frac{- 70 + 15}{6} + (6 \cdot 2) - 6 \cdot - 3$

Step 9.4.23.3

Add $- 70$ and $15$ .

$\frac{- 55}{6} + (6 \cdot 2) - 6 \cdot - 3$

Step 9.4.24

Move the negative in front of the fraction.

$- \frac{55}{6} + (6 \cdot 2) - 6 \cdot - 3$

Step 9.4.25

Multiply $6$ by $2$ .

$- \frac{55}{6} + 12 - 6 \cdot - 3$

Step 9.4.26

Multiply $- 6$ by $- 3$ .

$- \frac{55}{6} + 12 + 18$

Step 9.4.27

Add $12$ and $18$ .

$- \frac{55}{6} + 30$

Step 9.4.28

To write $30$ as a fraction with a common denominator, multiply by $\frac{6}{6}$ .

$- \frac{55}{6} + 30 \cdot \frac{6}{6}$

Step 9.4.29

Combine $30$ and $\frac{6}{6}$ .

$- \frac{55}{6} + \frac{30 \cdot 6}{6}$

Step 9.4.30

Combine the numerators over the common denominator.

$\frac{- 55 + 30 \cdot 6}{6}$

Step 9.4.31

Simplify the numerator.

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Step 9.4.31.1

Multiply $30$ by $6$ .

$\frac{- 55 + 180}{6}$

Step 9.4.31.2

Add $- 55$ and $180$ .

$\frac{125}{6}$

Step 10

The result can be shown in multiple forms.

Exact Form:

$\frac{125}{6}$

Decimal Form:

$20.8\overline{3}$

Mixed Number Form:

$20 \frac{5}{6}$

Step 11