Calculus Examples

$\int_{- 1}^{5} [4 y - (y^{2} - 5)] d y$

Step 1

Remove parentheses.

$\int_{- 1}^{5} 4 y - (y^{2} - 5) d y$

Step 2

Split the single integral into multiple integrals.

$\int_{- 1}^{5} 4 y d y + \int_{- 1}^{5} - (y^{2} - 5) d y$

Step 3

Since $4$ is constant with respect to $y$ , move $4$ out of the integral.

$4 \int_{- 1}^{5} y d y + \int_{- 1}^{5} - (y^{2} - 5) d y$

Step 4

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

$4 (\frac{1}{2} y^{2}]_{- 1}^{5}) + \int_{- 1}^{5} - (y^{2} - 5) d y$

Step 5

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

$4 (\frac{y^{2}}{2}]_{- 1}^{5}) + \int_{- 1}^{5} - (y^{2} - 5) d y$

Step 6

Multiply $- (y^{2} - 5)$ .

$4 (\frac{y^{2}}{2}]_{- 1}^{5}) + \int_{- 1}^{5} - y^{2} - - 5 d y$

Step 7

Multiply $- 1$ by $- 5$ .

$4 (\frac{y^{2}}{2}]_{- 1}^{5}) + \int_{- 1}^{5} - y^{2} + 5 d y$

Step 8

Split the single integral into multiple integrals.

$4 (\frac{y^{2}}{2}]_{- 1}^{5}) + \int_{- 1}^{5} - y^{2} d y + \int_{- 1}^{5} 5 d y$

Step 9

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

$4 (\frac{y^{2}}{2}]_{- 1}^{5}) - \int_{- 1}^{5} y^{2} d y + \int_{- 1}^{5} 5 d y$

Step 10

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

$4 (\frac{y^{2}}{2}]_{- 1}^{5}) - (\frac{1}{3} y^{3}]_{- 1}^{5}) + \int_{- 1}^{5} 5 d y$

Step 11

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

$4 (\frac{y^{2}}{2}]_{- 1}^{5}) - (\frac{y^{3}}{3}]_{- 1}^{5}) + \int_{- 1}^{5} 5 d y$

Step 12

Apply the constant rule.

$4 (\frac{y^{2}}{2}]_{- 1}^{5}) - (\frac{y^{3}}{3}]_{- 1}^{5}) + 5 y]_{- 1}^{5}$

Step 13

Substitute and simplify.

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Step 13.1

Evaluate $\frac{y^{2}}{2}$ at $5$ and at $- 1$ .

$4 ((\frac{5^{2}}{2}) - \frac{{(- 1)}^{2}}{2}) - (\frac{y^{3}}{3}]_{- 1}^{5}) + 5 y]_{- 1}^{5}$

Step 13.2

Evaluate $\frac{y^{3}}{3}$ at $5$ and at $- 1$ .

$4 (\frac{5^{2}}{2} - \frac{{(- 1)}^{2}}{2}) - (\frac{5^{3}}{3} - \frac{{(- 1)}^{3}}{3}) + 5 y]_{- 1}^{5}$

Step 13.3

Evaluate $5 y$ at $5$ and at $- 1$ .

$4 (\frac{5^{2}}{2} - \frac{{(- 1)}^{2}}{2}) - (\frac{5^{3}}{3} - \frac{{(- 1)}^{3}}{3}) + (5 \cdot 5) - 5 \cdot - 1$

Step 13.4

Simplify.

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Step 13.4.1

Raise $5$ to the power of $2$ .

$4 (\frac{25}{2} - \frac{{(- 1)}^{2}}{2}) - (\frac{5^{3}}{3} - \frac{{(- 1)}^{3}}{3}) + (5 \cdot 5) - 5 \cdot - 1$

Step 13.4.2

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

$4 (\frac{25}{2} - \frac{1}{2}) - (\frac{5^{3}}{3} - \frac{{(- 1)}^{3}}{3}) + (5 \cdot 5) - 5 \cdot - 1$

Step 13.4.3

Combine the numerators over the common denominator.

$4 \frac{25 - 1}{2} - (\frac{5^{3}}{3} - \frac{{(- 1)}^{3}}{3}) + (5 \cdot 5) - 5 \cdot - 1$

Step 13.4.4

Subtract $1$ from $25$ .

$4 (\frac{24}{2}) - (\frac{5^{3}}{3} - \frac{{(- 1)}^{3}}{3}) + (5 \cdot 5) - 5 \cdot - 1$

Step 13.4.5

Cancel the common factor of $24$ and $2$ .

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Step 13.4.5.1

Factor $2$ out of $24$ .

$4 \frac{2 \cdot 12}{2} - (\frac{5^{3}}{3} - \frac{{(- 1)}^{3}}{3}) + (5 \cdot 5) - 5 \cdot - 1$

Step 13.4.5.2

Cancel the common factors.

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Step 13.4.5.2.1

Factor $2$ out of $2$ .

$4 \frac{2 \cdot 12}{2 (1)} - (\frac{5^{3}}{3} - \frac{{(- 1)}^{3}}{3}) + (5 \cdot 5) - 5 \cdot - 1$

Step 13.4.5.2.2

Cancel the common factor.

$4 \frac{2 \cdot 12}{2 \cdot 1} - (\frac{5^{3}}{3} - \frac{{(- 1)}^{3}}{3}) + (5 \cdot 5) - 5 \cdot - 1$

Step 13.4.5.2.3

Rewrite the expression.

$4 (\frac{12}{1}) - (\frac{5^{3}}{3} - \frac{{(- 1)}^{3}}{3}) + (5 \cdot 5) - 5 \cdot - 1$

Step 13.4.5.2.4

Divide $12$ by $1$ .

$4 \cdot 12 - (\frac{5^{3}}{3} - \frac{{(- 1)}^{3}}{3}) + (5 \cdot 5) - 5 \cdot - 1$

Step 13.4.6

Multiply $4$ by $12$ .

$48 - (\frac{5^{3}}{3} - \frac{{(- 1)}^{3}}{3}) + (5 \cdot 5) - 5 \cdot - 1$

Step 13.4.7

Raise $5$ to the power of $3$ .

$48 - (\frac{125}{3} - \frac{{(- 1)}^{3}}{3}) + (5 \cdot 5) - 5 \cdot - 1$

Step 13.4.8

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

$48 - (\frac{125}{3} - \frac{- 1}{3}) + (5 \cdot 5) - 5 \cdot - 1$

Step 13.4.9

Move the negative in front of the fraction.

$48 - (\frac{125}{3} - - \frac{1}{3}) + (5 \cdot 5) - 5 \cdot - 1$

Step 13.4.10

Multiply $- 1$ by $- 1$ .

$48 - (\frac{125}{3} + 1 (\frac{1}{3})) + (5 \cdot 5) - 5 \cdot - 1$

Step 13.4.11

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

$48 - (\frac{125}{3} + \frac{1}{3}) + (5 \cdot 5) - 5 \cdot - 1$

Step 13.4.12

Combine the numerators over the common denominator.

$48 - \frac{125 + 1}{3} + (5 \cdot 5) - 5 \cdot - 1$

Step 13.4.13

Add $125$ and $1$ .

$48 - \frac{126}{3} + (5 \cdot 5) - 5 \cdot - 1$

Step 13.4.14

Cancel the common factor of $126$ and $3$ .

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Step 13.4.14.1

Factor $3$ out of $126$ .

$48 - \frac{3 \cdot 42}{3} + (5 \cdot 5) - 5 \cdot - 1$

Step 13.4.14.2

Cancel the common factors.

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Step 13.4.14.2.1

Factor $3$ out of $3$ .

$48 - \frac{3 \cdot 42}{3 (1)} + (5 \cdot 5) - 5 \cdot - 1$

Step 13.4.14.2.2

Cancel the common factor.

$48 - \frac{3 \cdot 42}{3 \cdot 1} + (5 \cdot 5) - 5 \cdot - 1$

Step 13.4.14.2.3

Rewrite the expression.

$48 - \frac{42}{1} + (5 \cdot 5) - 5 \cdot - 1$

Step 13.4.14.2.4

Divide $42$ by $1$ .

$48 - 1 \cdot 42 + (5 \cdot 5) - 5 \cdot - 1$

Step 13.4.15

Multiply $- 1$ by $42$ .

$48 - 42 + (5 \cdot 5) - 5 \cdot - 1$

Step 13.4.16

Subtract $42$ from $48$ .

$6 + (5 \cdot 5) - 5 \cdot - 1$

Step 13.4.17

Multiply $5$ by $5$ .

$6 + 25 - 5 \cdot - 1$

Step 13.4.18

Multiply $- 5$ by $- 1$ .

$6 + 25 + 5$

Step 13.4.19

Add $25$ and $5$ .

$6 + 30$

Step 13.4.20

Add $6$ and $30$ .

$36$

Step 14