Calculus Examples

$\int_{- 2}^{3} [y - (y^{2} - 6)] d y$

Step 1

Remove parentheses.

$\int_{- 2}^{3} y - (y^{2} - 6) d y$

Step 2

Split the single integral into multiple integrals.

$\int_{- 2}^{3} y d y + \int_{- 2}^{3} - (y^{2} - 6) d y$

Step 3

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

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

Step 4

Multiply $- (y^{2} - 6)$ .

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

Step 5

Multiply $- 1$ by $- 6$ .

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

Step 6

Split the single integral into multiple integrals.

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

Step 7

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

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

Step 8

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

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

Step 9

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

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

Step 10

Apply the constant rule.

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

Step 11

Simplify the answer.

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Step 11.1

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

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

Step 11.2

Substitute and simplify.

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Step 11.2.1

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

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

Step 11.2.2

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

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

Step 11.2.3

Simplify.

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Step 11.2.3.1

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

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

Step 11.2.3.2

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

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

Step 11.2.3.3

Multiply $6$ by $3$ .

$\frac{9}{2} + 18 - (\frac{1}{2} {(- 2)}^{2} + 6 \cdot - 2) - ((\frac{3^{3}}{3}) - \frac{{(- 2)}^{3}}{3})$

Step 11.2.3.4

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

$\frac{9}{2} + 18 \cdot \frac{2}{2} - (\frac{1}{2} {(- 2)}^{2} + 6 \cdot - 2) - ((\frac{3^{3}}{3}) - \frac{{(- 2)}^{3}}{3})$

Step 11.2.3.5

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

$\frac{9}{2} + \frac{18 \cdot 2}{2} - (\frac{1}{2} {(- 2)}^{2} + 6 \cdot - 2) - ((\frac{3^{3}}{3}) - \frac{{(- 2)}^{3}}{3})$

Step 11.2.3.6

Combine the numerators over the common denominator.

$\frac{9 + 18 \cdot 2}{2} - (\frac{1}{2} {(- 2)}^{2} + 6 \cdot - 2) - ((\frac{3^{3}}{3}) - \frac{{(- 2)}^{3}}{3})$

Step 11.2.3.7

Simplify the numerator.

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Step 11.2.3.7.1

Multiply $18$ by $2$ .

$\frac{9 + 36}{2} - (\frac{1}{2} {(- 2)}^{2} + 6 \cdot - 2) - ((\frac{3^{3}}{3}) - \frac{{(- 2)}^{3}}{3})$

Step 11.2.3.7.2

Add $9$ and $36$ .

$\frac{45}{2} - (\frac{1}{2} {(- 2)}^{2} + 6 \cdot - 2) - ((\frac{3^{3}}{3}) - \frac{{(- 2)}^{3}}{3})$

Step 11.2.3.8

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

$\frac{45}{2} - (\frac{1}{2} \cdot 4 + 6 \cdot - 2) - ((\frac{3^{3}}{3}) - \frac{{(- 2)}^{3}}{3})$

Step 11.2.3.9

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

$\frac{45}{2} - (\frac{4}{2} + 6 \cdot - 2) - ((\frac{3^{3}}{3}) - \frac{{(- 2)}^{3}}{3})$

Step 11.2.3.10

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

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Step 11.2.3.10.1

Factor $2$ out of $4$ .

$\frac{45}{2} - (\frac{2 \cdot 2}{2} + 6 \cdot - 2) - ((\frac{3^{3}}{3}) - \frac{{(- 2)}^{3}}{3})$

Step 11.2.3.10.2

Cancel the common factors.

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Step 11.2.3.10.2.1

Factor $2$ out of $2$ .

$\frac{45}{2} - (\frac{2 \cdot 2}{2 (1)} + 6 \cdot - 2) - ((\frac{3^{3}}{3}) - \frac{{(- 2)}^{3}}{3})$

Step 11.2.3.10.2.2

Cancel the common factor.

$\frac{45}{2} - (\frac{2 \cdot 2}{2 \cdot 1} + 6 \cdot - 2) - ((\frac{3^{3}}{3}) - \frac{{(- 2)}^{3}}{3})$

Step 11.2.3.10.2.3

Rewrite the expression.

$\frac{45}{2} - (\frac{2}{1} + 6 \cdot - 2) - ((\frac{3^{3}}{3}) - \frac{{(- 2)}^{3}}{3})$

Step 11.2.3.10.2.4

Divide $2$ by $1$ .

$\frac{45}{2} - (2 + 6 \cdot - 2) - ((\frac{3^{3}}{3}) - \frac{{(- 2)}^{3}}{3})$

Step 11.2.3.11

Multiply $6$ by $- 2$ .

$\frac{45}{2} - (2 - 12) - ((\frac{3^{3}}{3}) - \frac{{(- 2)}^{3}}{3})$

Step 11.2.3.12

Subtract $12$ from $2$ .

$\frac{45}{2} - - 10 - ((\frac{3^{3}}{3}) - \frac{{(- 2)}^{3}}{3})$

Step 11.2.3.13

Multiply $- 1$ by $- 10$ .

$\frac{45}{2} + 10 - ((\frac{3^{3}}{3}) - \frac{{(- 2)}^{3}}{3})$

Step 11.2.3.14

To write $10$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$\frac{45}{2} + 10 \cdot \frac{2}{2} - ((\frac{3^{3}}{3}) - \frac{{(- 2)}^{3}}{3})$

Step 11.2.3.15

Combine $10$ and $\frac{2}{2}$ .

$\frac{45}{2} + \frac{10 \cdot 2}{2} - ((\frac{3^{3}}{3}) - \frac{{(- 2)}^{3}}{3})$

Step 11.2.3.16

Combine the numerators over the common denominator.

$\frac{45 + 10 \cdot 2}{2} - ((\frac{3^{3}}{3}) - \frac{{(- 2)}^{3}}{3})$

Step 11.2.3.17

Simplify the numerator.

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Step 11.2.3.17.1

Multiply $10$ by $2$ .

$\frac{45 + 20}{2} - ((\frac{3^{3}}{3}) - \frac{{(- 2)}^{3}}{3})$

Step 11.2.3.17.2

Add $45$ and $20$ .

$\frac{65}{2} - ((\frac{3^{3}}{3}) - \frac{{(- 2)}^{3}}{3})$

Step 11.2.3.18

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

$\frac{65}{2} - (\frac{27}{3} - \frac{{(- 2)}^{3}}{3})$

Step 11.2.3.19

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

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Step 11.2.3.19.1

Factor $3$ out of $27$ .

$\frac{65}{2} - (\frac{3 \cdot 9}{3} - \frac{{(- 2)}^{3}}{3})$

Step 11.2.3.19.2

Cancel the common factors.

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Step 11.2.3.19.2.1

Factor $3$ out of $3$ .

$\frac{65}{2} - (\frac{3 \cdot 9}{3 (1)} - \frac{{(- 2)}^{3}}{3})$

Step 11.2.3.19.2.2

Cancel the common factor.

$\frac{65}{2} - (\frac{3 \cdot 9}{3 \cdot 1} - \frac{{(- 2)}^{3}}{3})$

Step 11.2.3.19.2.3

Rewrite the expression.

$\frac{65}{2} - (\frac{9}{1} - \frac{{(- 2)}^{3}}{3})$

Step 11.2.3.19.2.4

Divide $9$ by $1$ .

$\frac{65}{2} - (9 - \frac{{(- 2)}^{3}}{3})$

Step 11.2.3.20

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

$\frac{65}{2} - (9 - \frac{- 8}{3})$

Step 11.2.3.21

Move the negative in front of the fraction.

$\frac{65}{2} - (9 - - \frac{8}{3})$

Step 11.2.3.22

Multiply $- 1$ by $- 1$ .

$\frac{65}{2} - (9 + 1 (\frac{8}{3}))$

Step 11.2.3.23

Multiply $\frac{8}{3}$ by $1$ .

$\frac{65}{2} - (9 + \frac{8}{3})$

Step 11.2.3.24

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

$\frac{65}{2} - (9 \cdot \frac{3}{3} + \frac{8}{3})$

Step 11.2.3.25

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

$\frac{65}{2} - (\frac{9 \cdot 3}{3} + \frac{8}{3})$

Step 11.2.3.26

Combine the numerators over the common denominator.

$\frac{65}{2} - \frac{9 \cdot 3 + 8}{3}$

Step 11.2.3.27

Simplify the numerator.

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Step 11.2.3.27.1

Multiply $9$ by $3$ .

$\frac{65}{2} - \frac{27 + 8}{3}$

Step 11.2.3.27.2

Add $27$ and $8$ .

$\frac{65}{2} - \frac{35}{3}$

Step 11.2.3.28

To write $\frac{65}{2}$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$\frac{65}{2} \cdot \frac{3}{3} - \frac{35}{3}$

Step 11.2.3.29

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

$\frac{65}{2} \cdot \frac{3}{3} - \frac{35}{3} \cdot \frac{2}{2}$

Step 11.2.3.30

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

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Step 11.2.3.30.1

Multiply $\frac{65}{2}$ by $\frac{3}{3}$ .

$\frac{65 \cdot 3}{2 \cdot 3} - \frac{35}{3} \cdot \frac{2}{2}$

Step 11.2.3.30.2

Multiply $2$ by $3$ .

$\frac{65 \cdot 3}{6} - \frac{35}{3} \cdot \frac{2}{2}$

Step 11.2.3.30.3

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

$\frac{65 \cdot 3}{6} - \frac{35 \cdot 2}{3 \cdot 2}$

Step 11.2.3.30.4

Multiply $3$ by $2$ .

$\frac{65 \cdot 3}{6} - \frac{35 \cdot 2}{6}$

Step 11.2.3.31

Combine the numerators over the common denominator.

$\frac{65 \cdot 3 - 35 \cdot 2}{6}$

Step 11.2.3.32

Simplify the numerator.

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Step 11.2.3.32.1

Multiply $65$ by $3$ .

$\frac{195 - 35 \cdot 2}{6}$

Step 11.2.3.32.2

Multiply $- 35$ by $2$ .

$\frac{195 - 70}{6}$

Step 11.2.3.32.3

Subtract $70$ from $195$ .

$\frac{125}{6}$

Step 12

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 13