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

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 3

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

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

Step 4

Apply the constant rule.

$\frac{1}{3} x^{3}]_{- 3}^{2} + \frac{1}{2} x^{2}]_{- 3}^{2} + - 6 x]_{- 3}^{2}$

Step 5

Simplify the answer.

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Step 5.1

Simplify.

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Step 5.1.1

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

$\frac{1}{3} x^{3} + \frac{1}{2} x^{2}]_{- 3}^{2} + - 6 x]_{- 3}^{2}$

Step 5.1.2

Combine $\frac{1}{3} x^{3} + \frac{1}{2} x^{2}]_{- 3}^{2}$ and $- 6 x]_{- 3}^{2}$ .

$\frac{1}{3} x^{3} + \frac{1}{2} x^{2} - 6 x]_{- 3}^{2}$

Step 5.2

Substitute and simplify.

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Step 5.2.1

Evaluate $\frac{1}{3} x^{3} + \frac{1}{2} x^{2} - 6 x$ at $2$ and at $- 3$ .

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

Step 5.2.2

Simplify.

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Step 5.2.2.1

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

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

Step 5.2.2.2

Combine $\frac{1}{3}$ and $8$ .

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

Step 5.2.2.3

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

$\frac{8}{3} + \frac{1}{2} \cdot 4 - 6 \cdot 2 - (\frac{1}{3} {(- 3)}^{3} + \frac{1}{2} {(- 3)}^{2} - 6 \cdot - 3)$

Step 5.2.2.4

Combine $\frac{1}{2}$ and $4$ .

$\frac{8}{3} + \frac{4}{2} - 6 \cdot 2 - (\frac{1}{3} {(- 3)}^{3} + \frac{1}{2} {(- 3)}^{2} - 6 \cdot - 3)$

Step 5.2.2.5

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

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Step 5.2.2.5.1

Factor $2$ out of $4$ .

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

Step 5.2.2.5.2

Cancel the common factors.

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Step 5.2.2.5.2.1

Factor $2$ out of $2$ .

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

Step 5.2.2.5.2.2

Cancel the common factor.

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

Step 5.2.2.5.2.3

Rewrite the expression.

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

Step 5.2.2.5.2.4

Divide $2$ by $1$ .

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

Step 5.2.2.6

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

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

Step 5.2.2.7

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

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

Step 5.2.2.8

Combine the numerators over the common denominator.

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

Step 5.2.2.9

Simplify the numerator.

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Step 5.2.2.9.1

Multiply $2$ by $3$ .

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

Step 5.2.2.9.2

Add $8$ and $6$ .

$\frac{14}{3} - 6 \cdot 2 - (\frac{1}{3} {(- 3)}^{3} + \frac{1}{2} {(- 3)}^{2} - 6 \cdot - 3)$

Step 5.2.2.10

Multiply $- 6$ by $2$ .

$\frac{14}{3} - 12 - (\frac{1}{3} {(- 3)}^{3} + \frac{1}{2} {(- 3)}^{2} - 6 \cdot - 3)$

Step 5.2.2.11

To write $- 12$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$\frac{14}{3} - 12 \cdot \frac{3}{3} - (\frac{1}{3} {(- 3)}^{3} + \frac{1}{2} {(- 3)}^{2} - 6 \cdot - 3)$

Step 5.2.2.12

Combine $- 12$ and $\frac{3}{3}$ .

$\frac{14}{3} + \frac{- 12 \cdot 3}{3} - (\frac{1}{3} {(- 3)}^{3} + \frac{1}{2} {(- 3)}^{2} - 6 \cdot - 3)$

Step 5.2.2.13

Combine the numerators over the common denominator.

$\frac{14 - 12 \cdot 3}{3} - (\frac{1}{3} {(- 3)}^{3} + \frac{1}{2} {(- 3)}^{2} - 6 \cdot - 3)$

Step 5.2.2.14

Simplify the numerator.

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Step 5.2.2.14.1

Multiply $- 12$ by $3$ .

$\frac{14 - 36}{3} - (\frac{1}{3} {(- 3)}^{3} + \frac{1}{2} {(- 3)}^{2} - 6 \cdot - 3)$

Step 5.2.2.14.2

Subtract $36$ from $14$ .

$\frac{- 22}{3} - (\frac{1}{3} {(- 3)}^{3} + \frac{1}{2} {(- 3)}^{2} - 6 \cdot - 3)$

Step 5.2.2.15

Move the negative in front of the fraction.

$- \frac{22}{3} - (\frac{1}{3} {(- 3)}^{3} + \frac{1}{2} {(- 3)}^{2} - 6 \cdot - 3)$

Step 5.2.2.16

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

$- \frac{22}{3} - (\frac{1}{3} \cdot - 27 + \frac{1}{2} {(- 3)}^{2} - 6 \cdot - 3)$

Step 5.2.2.17

Combine $\frac{1}{3}$ and $- 27$ .

$- \frac{22}{3} - (\frac{- 27}{3} + \frac{1}{2} {(- 3)}^{2} - 6 \cdot - 3)$

Step 5.2.2.18

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

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Step 5.2.2.18.1

Factor $3$ out of $- 27$ .

$- \frac{22}{3} - (\frac{3 \cdot - 9}{3} + \frac{1}{2} {(- 3)}^{2} - 6 \cdot - 3)$

Step 5.2.2.18.2

Cancel the common factors.

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Step 5.2.2.18.2.1

Factor $3$ out of $3$ .

$- \frac{22}{3} - (\frac{3 \cdot - 9}{3 (1)} + \frac{1}{2} {(- 3)}^{2} - 6 \cdot - 3)$

Step 5.2.2.18.2.2

Cancel the common factor.

$- \frac{22}{3} - (\frac{3 \cdot - 9}{3 \cdot 1} + \frac{1}{2} {(- 3)}^{2} - 6 \cdot - 3)$

Step 5.2.2.18.2.3

Rewrite the expression.

$- \frac{22}{3} - (\frac{- 9}{1} + \frac{1}{2} {(- 3)}^{2} - 6 \cdot - 3)$

Step 5.2.2.18.2.4

Divide $- 9$ by $1$ .

$- \frac{22}{3} - (- 9 + \frac{1}{2} {(- 3)}^{2} - 6 \cdot - 3)$

Step 5.2.2.19

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

$- \frac{22}{3} - (- 9 + \frac{1}{2} \cdot 9 - 6 \cdot - 3)$

Step 5.2.2.20

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

$- \frac{22}{3} - (- 9 + \frac{9}{2} - 6 \cdot - 3)$

Step 5.2.2.21

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

$- \frac{22}{3} - (- 9 \cdot \frac{2}{2} + \frac{9}{2} - 6 \cdot - 3)$

Step 5.2.2.22

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

$- \frac{22}{3} - (\frac{- 9 \cdot 2}{2} + \frac{9}{2} - 6 \cdot - 3)$

Step 5.2.2.23

Combine the numerators over the common denominator.

$- \frac{22}{3} - (\frac{- 9 \cdot 2 + 9}{2} - 6 \cdot - 3)$

Step 5.2.2.24

Simplify the numerator.

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Step 5.2.2.24.1

Multiply $- 9$ by $2$ .

$- \frac{22}{3} - (\frac{- 18 + 9}{2} - 6 \cdot - 3)$

Step 5.2.2.24.2

Add $- 18$ and $9$ .

$- \frac{22}{3} - (\frac{- 9}{2} - 6 \cdot - 3)$

Step 5.2.2.25

Move the negative in front of the fraction.

$- \frac{22}{3} - (- \frac{9}{2} - 6 \cdot - 3)$

Step 5.2.2.26

Multiply $- 6$ by $- 3$ .

$- \frac{22}{3} - (- \frac{9}{2} + 18)$

Step 5.2.2.27

To write $18$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$- \frac{22}{3} - (- \frac{9}{2} + 18 \cdot \frac{2}{2})$

Step 5.2.2.28

Combine $18$ and $\frac{2}{2}$ .

$- \frac{22}{3} - (- \frac{9}{2} + \frac{18 \cdot 2}{2})$

Step 5.2.2.29

Combine the numerators over the common denominator.

$- \frac{22}{3} - \frac{- 9 + 18 \cdot 2}{2}$

Step 5.2.2.30

Simplify the numerator.

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Step 5.2.2.30.1

Multiply $18$ by $2$ .

$- \frac{22}{3} - \frac{- 9 + 36}{2}$

Step 5.2.2.30.2

Add $- 9$ and $36$ .

$- \frac{22}{3} - \frac{27}{2}$

Step 5.2.2.31

To write $- \frac{22}{3}$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$- \frac{22}{3} \cdot \frac{2}{2} - \frac{27}{2}$

Step 5.2.2.32

To write $- \frac{27}{2}$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$- \frac{22}{3} \cdot \frac{2}{2} - \frac{27}{2} \cdot \frac{3}{3}$

Step 5.2.2.33

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

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Step 5.2.2.33.1

Multiply $\frac{22}{3}$ by $\frac{2}{2}$ .

$- \frac{22 \cdot 2}{3 \cdot 2} - \frac{27}{2} \cdot \frac{3}{3}$

Step 5.2.2.33.2

Multiply $3$ by $2$ .

$- \frac{22 \cdot 2}{6} - \frac{27}{2} \cdot \frac{3}{3}$

Step 5.2.2.33.3

Multiply $\frac{27}{2}$ by $\frac{3}{3}$ .

$- \frac{22 \cdot 2}{6} - \frac{27 \cdot 3}{2 \cdot 3}$

Step 5.2.2.33.4

Multiply $2$ by $3$ .

$- \frac{22 \cdot 2}{6} - \frac{27 \cdot 3}{6}$

Step 5.2.2.34

Combine the numerators over the common denominator.

$\frac{- 22 \cdot 2 - 27 \cdot 3}{6}$

Step 5.2.2.35

Simplify the numerator.

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Step 5.2.2.35.1

Multiply $- 22$ by $2$ .

$\frac{- 44 - 27 \cdot 3}{6}$

Step 5.2.2.35.2

Multiply $- 27$ by $3$ .

$\frac{- 44 - 81}{6}$

Step 5.2.2.35.3

Subtract $81$ from $- 44$ .

$\frac{- 125}{6}$

Step 5.2.2.36

Move the negative in front of the fraction.

$- \frac{125}{6}$

Step 6

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 7