Calculus Examples

$\int_{- 1}^{6} [5 y - (y^{2} - 6)] d y$

Step 1

Remove parentheses.

$\int_{- 1}^{6} 5 y - (y^{2} - 6) d y$

Step 2

Split the single integral into multiple integrals.

$\int_{- 1}^{6} 5 y d y + \int_{- 1}^{6} - (y^{2} - 6) d y$

Step 3

Since $5$ is constant with respect to $y$ , move $5$ out of the integral.

$5 \int_{- 1}^{6} y d y + \int_{- 1}^{6} - (y^{2} - 6) d y$

Step 4

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

$5 (\frac{1}{2} y^{2}]_{- 1}^{6}) + \int_{- 1}^{6} - (y^{2} - 6) d y$

Step 5

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

$5 (\frac{y^{2}}{2}]_{- 1}^{6}) + \int_{- 1}^{6} - (y^{2} - 6) d y$

Step 6

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

$5 (\frac{y^{2}}{2}]_{- 1}^{6}) + \int_{- 1}^{6} - y^{2} - - 6 d y$

Step 7

Multiply $- 1$ by $- 6$ .

$5 (\frac{y^{2}}{2}]_{- 1}^{6}) + \int_{- 1}^{6} - y^{2} + 6 d y$

Step 8

Split the single integral into multiple integrals.

$5 (\frac{y^{2}}{2}]_{- 1}^{6}) + \int_{- 1}^{6} - y^{2} d y + \int_{- 1}^{6} 6 d y$

Step 9

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

$5 (\frac{y^{2}}{2}]_{- 1}^{6}) - \int_{- 1}^{6} y^{2} d y + \int_{- 1}^{6} 6 d y$

Step 10

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

$5 (\frac{y^{2}}{2}]_{- 1}^{6}) - (\frac{1}{3} y^{3}]_{- 1}^{6}) + \int_{- 1}^{6} 6 d y$

Step 11

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

$5 (\frac{y^{2}}{2}]_{- 1}^{6}) - (\frac{y^{3}}{3}]_{- 1}^{6}) + \int_{- 1}^{6} 6 d y$

Step 12

Apply the constant rule.

$5 (\frac{y^{2}}{2}]_{- 1}^{6}) - (\frac{y^{3}}{3}]_{- 1}^{6}) + 6 y]_{- 1}^{6}$

Step 13

Substitute and simplify.

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Step 13.1

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

$5 ((\frac{6^{2}}{2}) - \frac{{(- 1)}^{2}}{2}) - (\frac{y^{3}}{3}]_{- 1}^{6}) + 6 y]_{- 1}^{6}$

Step 13.2

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

$5 (\frac{6^{2}}{2} - \frac{{(- 1)}^{2}}{2}) - (\frac{6^{3}}{3} - \frac{{(- 1)}^{3}}{3}) + 6 y]_{- 1}^{6}$

Step 13.3

Evaluate $6 y$ at $6$ and at $- 1$ .

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

Step 13.4

Simplify.

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Step 13.4.1

Raise $6$ to the power of $2$ .

$5 (\frac{36}{2} - \frac{{(- 1)}^{2}}{2}) - (\frac{6^{3}}{3} - \frac{{(- 1)}^{3}}{3}) + (6 \cdot 6) - 6 \cdot - 1$

Step 13.4.2

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

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Step 13.4.2.1

Factor $2$ out of $36$ .

$5 (\frac{2 \cdot 18}{2} - \frac{{(- 1)}^{2}}{2}) - (\frac{6^{3}}{3} - \frac{{(- 1)}^{3}}{3}) + (6 \cdot 6) - 6 \cdot - 1$

Step 13.4.2.2

Cancel the common factors.

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Step 13.4.2.2.1

Factor $2$ out of $2$ .

$5 (\frac{2 \cdot 18}{2 (1)} - \frac{{(- 1)}^{2}}{2}) - (\frac{6^{3}}{3} - \frac{{(- 1)}^{3}}{3}) + (6 \cdot 6) - 6 \cdot - 1$

Step 13.4.2.2.2

Cancel the common factor.

$5 (\frac{2 \cdot 18}{2 \cdot 1} - \frac{{(- 1)}^{2}}{2}) - (\frac{6^{3}}{3} - \frac{{(- 1)}^{3}}{3}) + (6 \cdot 6) - 6 \cdot - 1$

Step 13.4.2.2.3

Rewrite the expression.

$5 (\frac{18}{1} - \frac{{(- 1)}^{2}}{2}) - (\frac{6^{3}}{3} - \frac{{(- 1)}^{3}}{3}) + (6 \cdot 6) - 6 \cdot - 1$

Step 13.4.2.2.4

Divide $18$ by $1$ .

$5 (18 - \frac{{(- 1)}^{2}}{2}) - (\frac{6^{3}}{3} - \frac{{(- 1)}^{3}}{3}) + (6 \cdot 6) - 6 \cdot - 1$

Step 13.4.3

Raise $- 1$ to the power of $2$ .

$5 (18 - \frac{1}{2}) - (\frac{6^{3}}{3} - \frac{{(- 1)}^{3}}{3}) + (6 \cdot 6) - 6 \cdot - 1$

Step 13.4.4

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

$5 (18 \cdot \frac{2}{2} - \frac{1}{2}) - (\frac{6^{3}}{3} - \frac{{(- 1)}^{3}}{3}) + (6 \cdot 6) - 6 \cdot - 1$

Step 13.4.5

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

$5 (\frac{18 \cdot 2}{2} - \frac{1}{2}) - (\frac{6^{3}}{3} - \frac{{(- 1)}^{3}}{3}) + (6 \cdot 6) - 6 \cdot - 1$

Step 13.4.6

Combine the numerators over the common denominator.

$5 \frac{18 \cdot 2 - 1}{2} - (\frac{6^{3}}{3} - \frac{{(- 1)}^{3}}{3}) + (6 \cdot 6) - 6 \cdot - 1$

Step 13.4.7

Simplify the numerator.

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Step 13.4.7.1

Multiply $18$ by $2$ .

$5 \frac{36 - 1}{2} - (\frac{6^{3}}{3} - \frac{{(- 1)}^{3}}{3}) + (6 \cdot 6) - 6 \cdot - 1$

Step 13.4.7.2

Subtract $1$ from $36$ .

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

Step 13.4.8

Combine $5$ and $\frac{35}{2}$ .

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

Step 13.4.9

Multiply $5$ by $35$ .

$\frac{175}{2} - (\frac{6^{3}}{3} - \frac{{(- 1)}^{3}}{3}) + (6 \cdot 6) - 6 \cdot - 1$

Step 13.4.10

Raise $6$ to the power of $3$ .

$\frac{175}{2} - (\frac{216}{3} - \frac{{(- 1)}^{3}}{3}) + (6 \cdot 6) - 6 \cdot - 1$

Step 13.4.11

Cancel the common factor of $216$ and $3$ .

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Step 13.4.11.1

Factor $3$ out of $216$ .

$\frac{175}{2} - (\frac{3 \cdot 72}{3} - \frac{{(- 1)}^{3}}{3}) + (6 \cdot 6) - 6 \cdot - 1$

Step 13.4.11.2

Cancel the common factors.

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Step 13.4.11.2.1

Factor $3$ out of $3$ .

$\frac{175}{2} - (\frac{3 \cdot 72}{3 (1)} - \frac{{(- 1)}^{3}}{3}) + (6 \cdot 6) - 6 \cdot - 1$

Step 13.4.11.2.2

Cancel the common factor.

$\frac{175}{2} - (\frac{3 \cdot 72}{3 \cdot 1} - \frac{{(- 1)}^{3}}{3}) + (6 \cdot 6) - 6 \cdot - 1$

Step 13.4.11.2.3

Rewrite the expression.

$\frac{175}{2} - (\frac{72}{1} - \frac{{(- 1)}^{3}}{3}) + (6 \cdot 6) - 6 \cdot - 1$

Step 13.4.11.2.4

Divide $72$ by $1$ .

$\frac{175}{2} - (72 - \frac{{(- 1)}^{3}}{3}) + (6 \cdot 6) - 6 \cdot - 1$

Step 13.4.12

Raise $- 1$ to the power of $3$ .

$\frac{175}{2} - (72 - \frac{- 1}{3}) + (6 \cdot 6) - 6 \cdot - 1$

Step 13.4.13

Move the negative in front of the fraction.

$\frac{175}{2} - (72 - - \frac{1}{3}) + (6 \cdot 6) - 6 \cdot - 1$

Step 13.4.14

Multiply $- 1$ by $- 1$ .

$\frac{175}{2} - (72 + 1 (\frac{1}{3})) + (6 \cdot 6) - 6 \cdot - 1$

Step 13.4.15

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

$\frac{175}{2} - (72 + \frac{1}{3}) + (6 \cdot 6) - 6 \cdot - 1$

Step 13.4.16

To write $72$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$\frac{175}{2} - (72 \cdot \frac{3}{3} + \frac{1}{3}) + (6 \cdot 6) - 6 \cdot - 1$

Step 13.4.17

Combine $72$ and $\frac{3}{3}$ .

$\frac{175}{2} - (\frac{72 \cdot 3}{3} + \frac{1}{3}) + (6 \cdot 6) - 6 \cdot - 1$

Step 13.4.18

Combine the numerators over the common denominator.

$\frac{175}{2} - \frac{72 \cdot 3 + 1}{3} + (6 \cdot 6) - 6 \cdot - 1$

Step 13.4.19

Simplify the numerator.

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Step 13.4.19.1

Multiply $72$ by $3$ .

$\frac{175}{2} - \frac{216 + 1}{3} + (6 \cdot 6) - 6 \cdot - 1$

Step 13.4.19.2

Add $216$ and $1$ .

$\frac{175}{2} - \frac{217}{3} + (6 \cdot 6) - 6 \cdot - 1$

Step 13.4.20

To write $\frac{175}{2}$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$\frac{175}{2} \cdot \frac{3}{3} - \frac{217}{3} + (6 \cdot 6) - 6 \cdot - 1$

Step 13.4.21

To write $- \frac{217}{3}$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$\frac{175}{2} \cdot \frac{3}{3} - \frac{217}{3} \cdot \frac{2}{2} + (6 \cdot 6) - 6 \cdot - 1$

Step 13.4.22

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

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Step 13.4.22.1

Multiply $\frac{175}{2}$ by $\frac{3}{3}$ .

$\frac{175 \cdot 3}{2 \cdot 3} - \frac{217}{3} \cdot \frac{2}{2} + (6 \cdot 6) - 6 \cdot - 1$

Step 13.4.22.2

Multiply $2$ by $3$ .

$\frac{175 \cdot 3}{6} - \frac{217}{3} \cdot \frac{2}{2} + (6 \cdot 6) - 6 \cdot - 1$

Step 13.4.22.3

Multiply $\frac{217}{3}$ by $\frac{2}{2}$ .

$\frac{175 \cdot 3}{6} - \frac{217 \cdot 2}{3 \cdot 2} + (6 \cdot 6) - 6 \cdot - 1$

Step 13.4.22.4

Multiply $3$ by $2$ .

$\frac{175 \cdot 3}{6} - \frac{217 \cdot 2}{6} + (6 \cdot 6) - 6 \cdot - 1$

Step 13.4.23

Combine the numerators over the common denominator.

$\frac{175 \cdot 3 - 217 \cdot 2}{6} + (6 \cdot 6) - 6 \cdot - 1$

Step 13.4.24

Simplify the numerator.

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Step 13.4.24.1

Multiply $175$ by $3$ .

$\frac{525 - 217 \cdot 2}{6} + (6 \cdot 6) - 6 \cdot - 1$

Step 13.4.24.2

Multiply $- 217$ by $2$ .

$\frac{525 - 434}{6} + (6 \cdot 6) - 6 \cdot - 1$

Step 13.4.24.3

Subtract $434$ from $525$ .

$\frac{91}{6} + (6 \cdot 6) - 6 \cdot - 1$

Step 13.4.25

Multiply $6$ by $6$ .

$\frac{91}{6} + 36 - 6 \cdot - 1$

Step 13.4.26

Multiply $- 6$ by $- 1$ .

$\frac{91}{6} + 36 + 6$

Step 13.4.27

Add $36$ and $6$ .

$\frac{91}{6} + 42$

Step 13.4.28

To write $42$ as a fraction with a common denominator, multiply by $\frac{6}{6}$ .

$\frac{91}{6} + 42 \cdot \frac{6}{6}$

Step 13.4.29

Combine $42$ and $\frac{6}{6}$ .

$\frac{91}{6} + \frac{42 \cdot 6}{6}$

Step 13.4.30

Combine the numerators over the common denominator.

$\frac{91 + 42 \cdot 6}{6}$

Step 13.4.31

Simplify the numerator.

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Step 13.4.31.1

Multiply $42$ by $6$ .

$\frac{91 + 252}{6}$

Step 13.4.31.2

Add $91$ and $252$ .

$\frac{343}{6}$

Step 14

The result can be shown in multiple forms.

Exact Form:

$\frac{343}{6}$

Decimal Form:

$57.1\overline{6}$

Mixed Number Form:

$57 \frac{1}{6}$

Step 15