Calculus Examples

$\int_{1}^{5} {(7 - \frac{4}{y})}^{2} d y$

Step 1

Simplify.

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Step 1.1

Rewrite ${(7 - \frac{4}{y})}^{2}$ as $(7 - \frac{4}{y}) (7 - \frac{4}{y})$ .

$\int_{1}^{5} (7 - \frac{4}{y}) (7 - \frac{4}{y}) d y$

Step 1.2

Expand $(7 - \frac{4}{y}) (7 - \frac{4}{y})$ using the FOIL Method.

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Step 1.2.1

Apply the distributive property.

$\int_{1}^{5} 7 (7 - \frac{4}{y}) - \frac{4}{y} (7 - \frac{4}{y}) d y$

Step 1.2.2

Apply the distributive property.

$\int_{1}^{5} 7 \cdot 7 + 7 (- \frac{4}{y}) - \frac{4}{y} (7 - \frac{4}{y}) d y$

Step 1.2.3

Apply the distributive property.

$\int_{1}^{5} 7 \cdot 7 + 7 (- \frac{4}{y}) - \frac{4}{y} \cdot 7 - \frac{4}{y} (- \frac{4}{y}) d y$

Step 1.3

Simplify and combine like terms.

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Step 1.3.1

Simplify each term.

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Step 1.3.1.1

Multiply $7$ by $7$ .

$\int_{1}^{5} 49 + 7 (- \frac{4}{y}) - \frac{4}{y} \cdot 7 - \frac{4}{y} (- \frac{4}{y}) d y$

Step 1.3.1.2

Multiply $7 (- \frac{4}{y})$ .

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Step 1.3.1.2.1

Multiply $- 1$ by $7$ .

$\int_{1}^{5} 49 - 7 \frac{4}{y} - \frac{4}{y} \cdot 7 - \frac{4}{y} (- \frac{4}{y}) d y$

Step 1.3.1.2.2

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

$\int_{1}^{5} 49 + \frac{- 7 \cdot 4}{y} - \frac{4}{y} \cdot 7 - \frac{4}{y} (- \frac{4}{y}) d y$

Step 1.3.1.2.3

Multiply $- 7$ by $4$ .

$\int_{1}^{5} 49 + \frac{- 28}{y} - \frac{4}{y} \cdot 7 - \frac{4}{y} (- \frac{4}{y}) d y$

Step 1.3.1.3

Move the negative in front of the fraction.

$\int_{1}^{5} 49 - \frac{28}{y} - \frac{4}{y} \cdot 7 - \frac{4}{y} (- \frac{4}{y}) d y$

Step 1.3.1.4

Multiply $- \frac{4}{y} \cdot 7$ .

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Step 1.3.1.4.1

Multiply $7$ by $- 1$ .

$\int_{1}^{5} 49 - \frac{28}{y} - 7 \frac{4}{y} - \frac{4}{y} (- \frac{4}{y}) d y$

Step 1.3.1.4.2

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

$\int_{1}^{5} 49 - \frac{28}{y} + \frac{- 7 \cdot 4}{y} - \frac{4}{y} (- \frac{4}{y}) d y$

Step 1.3.1.4.3

Multiply $- 7$ by $4$ .

$\int_{1}^{5} 49 - \frac{28}{y} + \frac{- 28}{y} - \frac{4}{y} (- \frac{4}{y}) d y$

Step 1.3.1.5

Move the negative in front of the fraction.

$\int_{1}^{5} 49 - \frac{28}{y} - \frac{28}{y} - \frac{4}{y} (- \frac{4}{y}) d y$

Step 1.3.1.6

Multiply $- \frac{4}{y} (- \frac{4}{y})$ .

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Step 1.3.1.6.1

Multiply $- 1$ by $- 1$ .

$\int_{1}^{5} 49 - \frac{28}{y} - \frac{28}{y} + 1 \frac{4}{y} \frac{4}{y} d y$

Step 1.3.1.6.2

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

$\int_{1}^{5} 49 - \frac{28}{y} - \frac{28}{y} + \frac{4}{y} \cdot \frac{4}{y} d y$

Step 1.3.1.6.3

Multiply $\frac{4}{y}$ by $\frac{4}{y}$ .

$\int_{1}^{5} 49 - \frac{28}{y} - \frac{28}{y} + \frac{4 \cdot 4}{y \cdot y} d y$

Step 1.3.1.6.4

Multiply $4$ by $4$ .

$\int_{1}^{5} 49 - \frac{28}{y} - \frac{28}{y} + \frac{16}{y \cdot y} d y$

Step 1.3.1.6.5

Raise $y$ to the power of $1$ .

$\int_{1}^{5} 49 - \frac{28}{y} - \frac{28}{y} + \frac{16}{y^{1} y} d y$

Step 1.3.1.6.6

Raise $y$ to the power of $1$ .

$\int_{1}^{5} 49 - \frac{28}{y} - \frac{28}{y} + \frac{16}{y^{1} y^{1}} d y$

Step 1.3.1.6.7

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$\int_{1}^{5} 49 - \frac{28}{y} - \frac{28}{y} + \frac{16}{y^{1 + 1}} d y$

Step 1.3.1.6.8

Add $1$ and $1$ .

$\int_{1}^{5} 49 - \frac{28}{y} - \frac{28}{y} + \frac{16}{y^{2}} d y$

Step 1.3.2

Subtract $\frac{28}{y}$ from $- \frac{28}{y}$ .

$\int_{1}^{5} 49 - 2 \frac{28}{y} + \frac{16}{y^{2}} d y$

Step 1.4

Simplify each term.

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Step 1.4.1

Multiply $- 2 \frac{28}{y}$ .

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Step 1.4.1.1

Combine $- 2$ and $\frac{28}{y}$ .

$\int_{1}^{5} 49 + \frac{- 2 \cdot 28}{y} + \frac{16}{y^{2}} d y$

Step 1.4.1.2

Multiply $- 2$ by $28$ .

$\int_{1}^{5} 49 + \frac{- 56}{y} + \frac{16}{y^{2}} d y$

Step 1.4.2

Move the negative in front of the fraction.

$\int_{1}^{5} 49 - \frac{56}{y} + \frac{16}{y^{2}} d y$

Step 2

Split the single integral into multiple integrals.

$\int_{1}^{5} 49 d y + \int_{1}^{5} - \frac{56}{y} d y + \int_{1}^{5} \frac{16}{y^{2}} d y$

Step 3

Apply the constant rule.

$49 y]_{1}^{5} + \int_{1}^{5} - \frac{56}{y} d y + \int_{1}^{5} \frac{16}{y^{2}} d y$

Step 4

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

$49 y]_{1}^{5} - \int_{1}^{5} \frac{56}{y} d y + \int_{1}^{5} \frac{16}{y^{2}} d y$

Step 5

Since $56$ is constant with respect to $y$ , move $56$ out of the integral.

$49 y]_{1}^{5} - (56 \int_{1}^{5} \frac{1}{y} d y) + \int_{1}^{5} \frac{16}{y^{2}} d y$

Step 6

Multiply $56$ by $- 1$ .

$49 y]_{1}^{5} - 56 \int_{1}^{5} \frac{1}{y} d y + \int_{1}^{5} \frac{16}{y^{2}} d y$

Step 7

The integral of $\frac{1}{y}$ with respect to $y$ is $\ln (| y |)$ .

$49 y]_{1}^{5} - 56 (\ln (| y |)]_{1}^{5}) + \int_{1}^{5} \frac{16}{y^{2}} d y$

Step 8

Since $16$ is constant with respect to $y$ , move $16$ out of the integral.

$49 y]_{1}^{5} - 56 (\ln (| y |)]_{1}^{5}) + 16 \int_{1}^{5} \frac{1}{y^{2}} d y$

Step 9

Apply basic rules of exponents.

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Step 9.1

Move $y^{2}$ out of the denominator by raising it to the $- 1$ power.

$49 y]_{1}^{5} - 56 (\ln (| y |)]_{1}^{5}) + 16 \int_{1}^{5} {(y^{2})}^{- 1} d y$

Step 9.2

Multiply the exponents in ${(y^{2})}^{- 1}$ .

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Step 9.2.1

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$49 y]_{1}^{5} - 56 (\ln (| y |)]_{1}^{5}) + 16 \int_{1}^{5} y^{2 \cdot - 1} d y$

Step 9.2.2

Multiply $2$ by $- 1$ .

$49 y]_{1}^{5} - 56 (\ln (| y |)]_{1}^{5}) + 16 \int_{1}^{5} y^{- 2} d y$

Step 10

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

$49 y]_{1}^{5} - 56 (\ln (| y |)]_{1}^{5}) + 16 (- y^{- 1}]_{1}^{5})$

Step 11

Simplify the answer.

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Step 11.1

Substitute and simplify.

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Step 11.1.1

Evaluate $49 y$ at $5$ and at $1$ .

$(49 \cdot 5) - 49 \cdot 1 - 56 (\ln (| y |)]_{1}^{5}) + 16 (- y^{- 1}]_{1}^{5})$

Step 11.1.2

Evaluate $\ln (| y |)$ at $5$ and at $1$ .

$49 \cdot 5 - 49 \cdot 1 - 56 (\ln (| 5 |) - \ln (| 1 |)) + 16 (- y^{- 1}]_{1}^{5})$

Step 11.1.3

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

$49 \cdot 5 - 49 \cdot 1 - 56 (\ln (| 5 |) - \ln (| 1 |)) + 16 ((- 5^{- 1}) + 1^{- 1})$

Step 11.1.4

Simplify.

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Step 11.1.4.1

Multiply $49$ by $5$ .

$245 - 49 \cdot 1 - 56 (\ln (| 5 |) - \ln (| 1 |)) + 16 ((- 5^{- 1}) + 1^{- 1})$

Step 11.1.4.2

Multiply $- 49$ by $1$ .

$245 - 49 - 56 (\ln (| 5 |) - \ln (| 1 |)) + 16 ((- 5^{- 1}) + 1^{- 1})$

Step 11.1.4.3

Subtract $49$ from $245$ .

$196 - 56 (\ln (| 5 |) - \ln (| 1 |)) + 16 ((- 5^{- 1}) + 1^{- 1})$

Step 11.1.4.4

Rewrite the expression using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$196 - 56 (\ln (| 5 |) - \ln (| 1 |)) + 16 (- \frac{1}{5} + 1^{- 1})$

Step 11.1.4.5

One to any power is one.

$196 - 56 (\ln (| 5 |) - \ln (| 1 |)) + 16 (- \frac{1}{5} + 1)$

Step 11.1.4.6

Write $1$ as a fraction with a common denominator.

$196 - 56 (\ln (| 5 |) - \ln (| 1 |)) + 16 (- \frac{1}{5} + \frac{5}{5})$

Step 11.1.4.7

Combine the numerators over the common denominator.

$196 - 56 (\ln (| 5 |) - \ln (| 1 |)) + 16 \frac{- 1 + 5}{5}$

Step 11.1.4.8

Add $- 1$ and $5$ .

$196 - 56 (\ln (| 5 |) - \ln (| 1 |)) + 16 (\frac{4}{5})$

Step 11.1.4.9

Combine $16$ and $\frac{4}{5}$ .

$196 - 56 (\ln (| 5 |) - \ln (| 1 |)) + \frac{16 \cdot 4}{5}$

Step 11.1.4.10

Multiply $16$ by $4$ .

$196 - 56 (\ln (| 5 |) - \ln (| 1 |)) + \frac{64}{5}$

Step 11.1.4.11

To write $196$ as a fraction with a common denominator, multiply by $\frac{5}{5}$ .

$196 \cdot \frac{5}{5} + \frac{64}{5} - 56 (\ln (| 5 |) - \ln (| 1 |))$

Step 11.1.4.12

Combine $196$ and $\frac{5}{5}$ .

$\frac{196 \cdot 5}{5} + \frac{64}{5} - 56 (\ln (| 5 |) - \ln (| 1 |))$

Step 11.1.4.13

Combine the numerators over the common denominator.

$\frac{196 \cdot 5 + 64}{5} - 56 (\ln (| 5 |) - \ln (| 1 |))$

Step 11.1.4.14

Simplify the numerator.

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Step 11.1.4.14.1

Multiply $196$ by $5$ .

$\frac{980 + 64}{5} - 56 (\ln (| 5 |) - \ln (| 1 |))$

Step 11.1.4.14.2

Add $980$ and $64$ .

$\frac{1044}{5} - 56 (\ln (| 5 |) - \ln (| 1 |))$

Step 11.1.4.15

To write $- 56 (\ln (| 5 |) - \ln (| 1 |))$ as a fraction with a common denominator, multiply by $\frac{5}{5}$ .

$\frac{1044}{5} - 56 (\ln (| 5 |) - \ln (| 1 |)) \cdot \frac{5}{5}$

Step 11.1.4.16

Combine $- 56 (\ln (| 5 |) - \ln (| 1 |))$ and $\frac{5}{5}$ .

$\frac{1044}{5} + \frac{- 56 (\ln (| 5 |) - \ln (| 1 |)) \cdot 5}{5}$

Step 11.1.4.17

Combine the numerators over the common denominator.

$\frac{1044 - 56 (\ln (| 5 |) - \ln (| 1 |)) \cdot 5}{5}$

Step 11.1.4.18

Multiply $5$ by $- 56$ .

$\frac{1044 - 280 (\ln (| 5 |) - \ln (| 1 |))}{5}$

Step 11.2

Use the quotient property of logarithms, $\log_{b} (x) - \log_{b} (y) = \log_{b} (\frac{x}{y})$ .

$\frac{1044 - 280 \ln (\frac{| 5 |}{| 1 |})}{5}$

Step 11.3

Simplify.

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Step 11.3.1

The absolute value is the distance between a number and zero. The distance between $0$ and $5$ is $5$ .

$\frac{1044 - 280 \ln (\frac{5}{| 1 |})}{5}$

Step 11.3.2

The absolute value is the distance between a number and zero. The distance between $0$ and $1$ is $1$ .

$\frac{1044 - 280 \ln (\frac{5}{1})}{5}$

Step 11.3.3

Divide $5$ by $1$ .

$\frac{1044 - 280 \ln (5)}{5}$

Step 12

The result can be shown in multiple forms.

Exact Form:

$\frac{1044 - 280 \ln (5)}{5}$

Decimal Form:

$118.67147690 \dots$

Step 13