Calculus Examples

$\int_{- 2}^{- 1} 7 y^{3} + 4 y^{- 3} d y$

Step 1

Split the single integral into multiple integrals.

$\int_{- 2}^{- 1} 7 y^{3} d y + \int_{- 2}^{- 1} 4 y^{- 3} d y$

Step 2

Since $7$ is constant with respect to $y$ , move $7$ out of the integral.

$7 \int_{- 2}^{- 1} y^{3} d y + \int_{- 2}^{- 1} 4 y^{- 3} d y$

Step 3

By the Power Rule, the integral of $y^{3}$ with respect to $y$ is $\frac{1}{4} y^{4}$ .

$7 (\frac{1}{4} y^{4}]_{- 2}^{- 1}) + \int_{- 2}^{- 1} 4 y^{- 3} d y$

Step 4

Combine $\frac{1}{4}$ and $y^{4}$ .

$7 (\frac{y^{4}}{4}]_{- 2}^{- 1}) + \int_{- 2}^{- 1} 4 y^{- 3} d y$

Step 5

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

$7 (\frac{y^{4}}{4}]_{- 2}^{- 1}) + 4 \int_{- 2}^{- 1} y^{- 3} d y$

Step 6

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

$7 (\frac{y^{4}}{4}]_{- 2}^{- 1}) + 4 (- \frac{1}{2} y^{- 2}]_{- 2}^{- 1})$

Step 7

Simplify the answer.

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Step 7.1

Simplify.

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Step 7.1.1

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

$7 (\frac{y^{4}}{4}]_{- 2}^{- 1}) + 4 (- \frac{y^{- 2}}{2}]_{- 2}^{- 1})$

Step 7.1.2

Move $y^{- 2}$ to the denominator using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$7 (\frac{y^{4}}{4}]_{- 2}^{- 1}) + 4 (- \frac{1}{2 y^{2}}]_{- 2}^{- 1})$

Step 7.2

Substitute and simplify.

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Step 7.2.1

Evaluate $\frac{y^{4}}{4}$ at $- 1$ and at $- 2$ .

$7 ((\frac{{(- 1)}^{4}}{4}) - \frac{{(- 2)}^{4}}{4}) + 4 (- \frac{1}{2 y^{2}}]_{- 2}^{- 1})$

Step 7.2.2

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

$7 ((\frac{{(- 1)}^{4}}{4}) - \frac{{(- 2)}^{4}}{4}) + 4 ((- \frac{1}{2 {(- 1)}^{2}}) + \frac{1}{2 {(- 2)}^{2}})$

Step 7.2.3

Simplify.

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Step 7.2.3.1

Raise $- 1$ to the power of $4$ .

$7 (\frac{1}{4} - \frac{{(- 2)}^{4}}{4}) + 4 ((- \frac{1}{2 {(- 1)}^{2}}) + \frac{1}{2 {(- 2)}^{2}})$

Step 7.2.3.2

Raise $- 2$ to the power of $4$ .

$7 (\frac{1}{4} - \frac{16}{4}) + 4 ((- \frac{1}{2 {(- 1)}^{2}}) + \frac{1}{2 {(- 2)}^{2}})$

Step 7.2.3.3

Cancel the common factor of $16$ and $4$ .

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Step 7.2.3.3.1

Factor $4$ out of $16$ .

$7 (\frac{1}{4} - \frac{4 \cdot 4}{4}) + 4 ((- \frac{1}{2 {(- 1)}^{2}}) + \frac{1}{2 {(- 2)}^{2}})$

Step 7.2.3.3.2

Cancel the common factors.

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Step 7.2.3.3.2.1

Factor $4$ out of $4$ .

$7 (\frac{1}{4} - \frac{4 \cdot 4}{4 (1)}) + 4 ((- \frac{1}{2 {(- 1)}^{2}}) + \frac{1}{2 {(- 2)}^{2}})$

Step 7.2.3.3.2.2

Cancel the common factor.

$7 (\frac{1}{4} - \frac{4 \cdot 4}{4 \cdot 1}) + 4 ((- \frac{1}{2 {(- 1)}^{2}}) + \frac{1}{2 {(- 2)}^{2}})$

Step 7.2.3.3.2.3

Rewrite the expression.

$7 (\frac{1}{4} - \frac{4}{1}) + 4 ((- \frac{1}{2 {(- 1)}^{2}}) + \frac{1}{2 {(- 2)}^{2}})$

Step 7.2.3.3.2.4

Divide $4$ by $1$ .

$7 (\frac{1}{4} - 1 \cdot 4) + 4 ((- \frac{1}{2 {(- 1)}^{2}}) + \frac{1}{2 {(- 2)}^{2}})$

Step 7.2.3.4

Multiply $- 1$ by $4$ .

$7 (\frac{1}{4} - 4) + 4 ((- \frac{1}{2 {(- 1)}^{2}}) + \frac{1}{2 {(- 2)}^{2}})$

Step 7.2.3.5

To write $- 4$ as a fraction with a common denominator, multiply by $\frac{4}{4}$ .

$7 (\frac{1}{4} - 4 \cdot \frac{4}{4}) + 4 ((- \frac{1}{2 {(- 1)}^{2}}) + \frac{1}{2 {(- 2)}^{2}})$

Step 7.2.3.6

Combine $- 4$ and $\frac{4}{4}$ .

$7 (\frac{1}{4} + \frac{- 4 \cdot 4}{4}) + 4 ((- \frac{1}{2 {(- 1)}^{2}}) + \frac{1}{2 {(- 2)}^{2}})$

Step 7.2.3.7

Combine the numerators over the common denominator.

$7 \frac{1 - 4 \cdot 4}{4} + 4 ((- \frac{1}{2 {(- 1)}^{2}}) + \frac{1}{2 {(- 2)}^{2}})$

Step 7.2.3.8

Simplify the numerator.

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Step 7.2.3.8.1

Multiply $- 4$ by $4$ .

$7 \frac{1 - 16}{4} + 4 ((- \frac{1}{2 {(- 1)}^{2}}) + \frac{1}{2 {(- 2)}^{2}})$

Step 7.2.3.8.2

Subtract $16$ from $1$ .

$7 (\frac{- 15}{4}) + 4 ((- \frac{1}{2 {(- 1)}^{2}}) + \frac{1}{2 {(- 2)}^{2}})$

Step 7.2.3.9

Move the negative in front of the fraction.

$7 (- \frac{15}{4}) + 4 ((- \frac{1}{2 {(- 1)}^{2}}) + \frac{1}{2 {(- 2)}^{2}})$

Step 7.2.3.10

Multiply $- 1$ by $7$ .

$- 7 (\frac{15}{4}) + 4 ((- \frac{1}{2 {(- 1)}^{2}}) + \frac{1}{2 {(- 2)}^{2}})$

Step 7.2.3.11

Combine $- 7$ and $\frac{15}{4}$ .

$\frac{- 7 \cdot 15}{4} + 4 ((- \frac{1}{2 {(- 1)}^{2}}) + \frac{1}{2 {(- 2)}^{2}})$

Step 7.2.3.12

Multiply $- 7$ by $15$ .

$\frac{- 105}{4} + 4 ((- \frac{1}{2 {(- 1)}^{2}}) + \frac{1}{2 {(- 2)}^{2}})$

Step 7.2.3.13

Move the negative in front of the fraction.

$- \frac{105}{4} + 4 ((- \frac{1}{2 {(- 1)}^{2}}) + \frac{1}{2 {(- 2)}^{2}})$

Step 7.2.3.14

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

$- \frac{105}{4} + 4 (- \frac{1}{2 \cdot 1} + \frac{1}{2 {(- 2)}^{2}})$

Step 7.2.3.15

Multiply $2$ by $1$ .

$- \frac{105}{4} + 4 (- \frac{1}{2} + \frac{1}{2 {(- 2)}^{2}})$

Step 7.2.3.16

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

$- \frac{105}{4} + 4 (- \frac{1}{2} + \frac{1}{2 \cdot 4})$

Step 7.2.3.17

Multiply $2$ by $4$ .

$- \frac{105}{4} + 4 (- \frac{1}{2} + \frac{1}{8})$

Step 7.2.3.18

To write $- \frac{1}{2}$ as a fraction with a common denominator, multiply by $\frac{4}{4}$ .

$- \frac{105}{4} + 4 (- \frac{1}{2} \cdot \frac{4}{4} + \frac{1}{8})$

Step 7.2.3.19

Write each expression with a common denominator of $8$ , by multiplying each by an appropriate factor of $1$ .

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Step 7.2.3.19.1

Multiply $\frac{1}{2}$ by $\frac{4}{4}$ .

$- \frac{105}{4} + 4 (- \frac{4}{2 \cdot 4} + \frac{1}{8})$

Step 7.2.3.19.2

Multiply $2$ by $4$ .

$- \frac{105}{4} + 4 (- \frac{4}{8} + \frac{1}{8})$

Step 7.2.3.20

Combine the numerators over the common denominator.

$- \frac{105}{4} + 4 \frac{- 4 + 1}{8}$

Step 7.2.3.21

Add $- 4$ and $1$ .

$- \frac{105}{4} + 4 (\frac{- 3}{8})$

Step 7.2.3.22

Move the negative in front of the fraction.

$- \frac{105}{4} + 4 (- \frac{3}{8})$

Step 7.2.3.23

Multiply $- 1$ by $4$ .

$- \frac{105}{4} - 4 (\frac{3}{8})$

Step 7.2.3.24

Combine $- 4$ and $\frac{3}{8}$ .

$- \frac{105}{4} + \frac{- 4 \cdot 3}{8}$

Step 7.2.3.25

Multiply $- 4$ by $3$ .

$- \frac{105}{4} + \frac{- 12}{8}$

Step 7.2.3.26

Cancel the common factor of $- 12$ and $8$ .

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Step 7.2.3.26.1

Factor $4$ out of $- 12$ .

$- \frac{105}{4} + \frac{4 (- 3)}{8}$

Step 7.2.3.26.2

Cancel the common factors.

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Step 7.2.3.26.2.1

Factor $4$ out of $8$ .

$- \frac{105}{4} + \frac{4 \cdot - 3}{4 \cdot 2}$

Step 7.2.3.26.2.2

Cancel the common factor.

$- \frac{105}{4} + \frac{4 \cdot - 3}{4 \cdot 2}$

Step 7.2.3.26.2.3

Rewrite the expression.

$- \frac{105}{4} + \frac{- 3}{2}$

Step 7.2.3.27

Move the negative in front of the fraction.

$- \frac{105}{4} - \frac{3}{2}$

Step 7.2.3.28

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

$- \frac{105}{4} - \frac{3}{2} \cdot \frac{2}{2}$

Step 7.2.3.29

Write each expression with a common denominator of $4$ , by multiplying each by an appropriate factor of $1$ .

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Step 7.2.3.29.1

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

$- \frac{105}{4} - \frac{3 \cdot 2}{2 \cdot 2}$

Step 7.2.3.29.2

Multiply $2$ by $2$ .

$- \frac{105}{4} - \frac{3 \cdot 2}{4}$

Step 7.2.3.30

Combine the numerators over the common denominator.

$\frac{- 105 - 3 \cdot 2}{4}$

Step 7.2.3.31

Simplify the numerator.

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Step 7.2.3.31.1

Multiply $- 3$ by $2$ .

$\frac{- 105 - 6}{4}$

Step 7.2.3.31.2

Subtract $6$ from $- 105$ .

$\frac{- 111}{4}$

Step 7.2.3.32

Move the negative in front of the fraction.

$- \frac{111}{4}$

Step 8

The result can be shown in multiple forms.

Exact Form:

$- \frac{111}{4}$

Decimal Form:

$- 27.75$

Mixed Number Form:

$- 27 \frac{3}{4}$

Step 9