Calculus Examples

$\int_{64}^{8} 1 \frac{\ln (y)}{\sqrt{y}} d y$

Step 1

Multiply $\frac{\ln (y)}{\sqrt{y}}$ by $1$ .

$\int_{64}^{8} \frac{\ln (y)}{\sqrt{y}} d y$

Step 2

Rewrite $\frac{\ln (y)}{\sqrt{y}}$ as a product.

$\int_{64}^{8} \ln (y) \frac{1}{\sqrt{y}} d y$

Step 3

Integrate by parts using the formula $\int u d v = u v - \int v d u$ , where $u = \ln (y)$ and $d v = \frac{1}{\sqrt{y}}$ .

$\ln (y) (2 y^{\frac{1}{2}})]_{64}^{8} - \int_{64}^{8} 2 y^{\frac{1}{2}} \frac{1}{y} d y$

Step 4

Simplify.

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Step 4.1

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

$\ln (y) (2 y^{\frac{1}{2}})]_{64}^{8} - \int_{64}^{8} y^{\frac{1}{2}} \frac{2}{y} d y$

Step 4.2

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

$\ln (y) (2 y^{\frac{1}{2}})]_{64}^{8} - \int_{64}^{8} \frac{y^{\frac{1}{2}} \cdot 2}{y} d y$

Step 4.3

Move $2$ to the left of $y^{\frac{1}{2}}$ .

$\ln (y) (2 y^{\frac{1}{2}})]_{64}^{8} - \int_{64}^{8} \frac{2 \cdot y^{\frac{1}{2}}}{y} d y$

Step 4.4

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

$\ln (y) (2 y^{\frac{1}{2}})]_{64}^{8} - \int_{64}^{8} \frac{2}{y \cdot y^{- \frac{1}{2}}} d y$

Step 4.5

Multiply $y$ by $y^{- \frac{1}{2}}$ by adding the exponents.

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Step 4.5.1

Multiply $y$ by $y^{- \frac{1}{2}}$ .

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Step 4.5.1.1

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

$\ln (y) (2 y^{\frac{1}{2}})]_{64}^{8} - \int_{64}^{8} \frac{2}{y^{1} y^{- \frac{1}{2}}} d y$

Step 4.5.1.2

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

$\ln (y) (2 y^{\frac{1}{2}})]_{64}^{8} - \int_{64}^{8} \frac{2}{y^{1 - \frac{1}{2}}} d y$

Step 4.5.2

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

$\ln (y) (2 y^{\frac{1}{2}})]_{64}^{8} - \int_{64}^{8} \frac{2}{y^{\frac{2}{2} - \frac{1}{2}}} d y$

Step 4.5.3

Combine the numerators over the common denominator.

$\ln (y) (2 y^{\frac{1}{2}})]_{64}^{8} - \int_{64}^{8} \frac{2}{y^{\frac{2 - 1}{2}}} d y$

Step 4.5.4

Subtract $1$ from $2$ .

$\ln (y) (2 y^{\frac{1}{2}})]_{64}^{8} - \int_{64}^{8} \frac{2}{y^{\frac{1}{2}}} d y$

Step 5

Since $2$ is constant with respect to $y$ , move $2$ out of the integral.

$\ln (y) (2 y^{\frac{1}{2}})]_{64}^{8} - (2 \int_{64}^{8} \frac{1}{y^{\frac{1}{2}}} d y)$

Step 6

Simplify the expression.

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Step 6.1

Multiply $2$ by $- 1$ .

$\ln (y) (2 y^{\frac{1}{2}})]_{64}^{8} - 2 \int_{64}^{8} \frac{1}{y^{\frac{1}{2}}} d y$

Step 6.2

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

$\ln (y) (2 y^{\frac{1}{2}})]_{64}^{8} - 2 \int_{64}^{8} {(y^{\frac{1}{2}})}^{- 1} d y$

Step 6.3

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

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Step 6.3.1

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

$\ln (y) (2 y^{\frac{1}{2}})]_{64}^{8} - 2 \int_{64}^{8} y^{\frac{1}{2} \cdot - 1} d y$

Step 6.3.2

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

$\ln (y) (2 y^{\frac{1}{2}})]_{64}^{8} - 2 \int_{64}^{8} y^{\frac{- 1}{2}} d y$

Step 6.3.3

Move the negative in front of the fraction.

$\ln (y) (2 y^{\frac{1}{2}})]_{64}^{8} - 2 \int_{64}^{8} y^{- \frac{1}{2}} d y$

Step 7

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

$\ln (y) (2 y^{\frac{1}{2}})]_{64}^{8} - 2 (2 y^{\frac{1}{2}}]_{64}^{8})$

Step 8

Substitute and simplify.

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Step 8.1

Evaluate $\ln (y) (2 y^{\frac{1}{2}})$ at $8$ and at $64$ .

$(\ln (8) (2 \cdot 8^{\frac{1}{2}})) - \ln (64) (2 \cdot 64^{\frac{1}{2}}) - 2 (2 y^{\frac{1}{2}}]_{64}^{8})$

Step 8.2

Evaluate $2 y^{\frac{1}{2}}$ at $8$ and at $64$ .

$(\ln (8) (2 \cdot 8^{\frac{1}{2}})) - \ln (64) (2 \cdot 64^{\frac{1}{2}}) - 2 ((2 \cdot 8^{\frac{1}{2}}) - 2 \cdot 64^{\frac{1}{2}})$

Step 8.3

Simplify.

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Step 8.3.1

Rewrite $8$ as $2^{3}$ .

$\ln (8) (2 \cdot {(2^{3})}^{\frac{1}{2}}) - \ln (64) (2 \cdot 64^{\frac{1}{2}}) - 2 ((2 \cdot 8^{\frac{1}{2}}) - 2 \cdot 64^{\frac{1}{2}})$

Step 8.3.2

Multiply the exponents in ${(2^{3})}^{\frac{1}{2}}$ .

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Step 8.3.2.1

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

$\ln (8) (2 \cdot 2^{3 (\frac{1}{2})}) - \ln (64) (2 \cdot 64^{\frac{1}{2}}) - 2 ((2 \cdot 8^{\frac{1}{2}}) - 2 \cdot 64^{\frac{1}{2}})$

Step 8.3.2.2

Combine $3$ and $\frac{1}{2}$ .

$\ln (8) (2 \cdot 2^{\frac{3}{2}}) - \ln (64) (2 \cdot 64^{\frac{1}{2}}) - 2 ((2 \cdot 8^{\frac{1}{2}}) - 2 \cdot 64^{\frac{1}{2}})$

Step 8.3.3

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

$\ln (8) \cdot 2^{1 + \frac{3}{2}} - \ln (64) (2 \cdot 64^{\frac{1}{2}}) - 2 ((2 \cdot 8^{\frac{1}{2}}) - 2 \cdot 64^{\frac{1}{2}})$

Step 8.3.4

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

$\ln (8) \cdot 2^{\frac{2}{2} + \frac{3}{2}} - \ln (64) (2 \cdot 64^{\frac{1}{2}}) - 2 ((2 \cdot 8^{\frac{1}{2}}) - 2 \cdot 64^{\frac{1}{2}})$

Step 8.3.5

Combine the numerators over the common denominator.

$\ln (8) \cdot 2^{\frac{2 + 3}{2}} - \ln (64) (2 \cdot 64^{\frac{1}{2}}) - 2 ((2 \cdot 8^{\frac{1}{2}}) - 2 \cdot 64^{\frac{1}{2}})$

Step 8.3.6

Add $2$ and $3$ .

$\ln (8) \cdot 2^{\frac{5}{2}} - \ln (64) (2 \cdot 64^{\frac{1}{2}}) - 2 ((2 \cdot 8^{\frac{1}{2}}) - 2 \cdot 64^{\frac{1}{2}})$

Step 8.3.7

Rewrite $64$ as $8^{2}$ .

$\ln (8) \cdot 2^{\frac{5}{2}} - \ln (64) (2 \cdot {(8^{2})}^{\frac{1}{2}}) - 2 ((2 \cdot 8^{\frac{1}{2}}) - 2 \cdot 64^{\frac{1}{2}})$

Step 8.3.8

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

$\ln (8) \cdot 2^{\frac{5}{2}} - \ln (64) (2 \cdot 8^{2 (\frac{1}{2})}) - 2 ((2 \cdot 8^{\frac{1}{2}}) - 2 \cdot 64^{\frac{1}{2}})$

Step 8.3.9

Cancel the common factor of $2$ .

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Step 8.3.9.1

Cancel the common factor.

$\ln (8) \cdot 2^{\frac{5}{2}} - \ln (64) (2 \cdot 8^{2 (\frac{1}{2})}) - 2 ((2 \cdot 8^{\frac{1}{2}}) - 2 \cdot 64^{\frac{1}{2}})$

Step 8.3.9.2

Rewrite the expression.

$\ln (8) \cdot 2^{\frac{5}{2}} - \ln (64) (2 \cdot 8^{1}) - 2 ((2 \cdot 8^{\frac{1}{2}}) - 2 \cdot 64^{\frac{1}{2}})$

Step 8.3.10

Evaluate the exponent.

$\ln (8) \cdot 2^{\frac{5}{2}} - \ln (64) (2 \cdot 8) - 2 ((2 \cdot 8^{\frac{1}{2}}) - 2 \cdot 64^{\frac{1}{2}})$

Step 8.3.11

Multiply $2$ by $8$ .

$\ln (8) \cdot 2^{\frac{5}{2}} - \ln (64) \cdot 16 - 2 ((2 \cdot 8^{\frac{1}{2}}) - 2 \cdot 64^{\frac{1}{2}})$

Step 8.3.12

Multiply $16$ by $- 1$ .

$\ln (8) \cdot 2^{\frac{5}{2}} - 16 \ln (64) - 2 ((2 \cdot 8^{\frac{1}{2}}) - 2 \cdot 64^{\frac{1}{2}})$

Step 8.3.13

Rewrite $8$ as $2^{3}$ .

$\ln (8) \cdot 2^{\frac{5}{2}} - 16 \ln (64) - 2 (2 \cdot {(2^{3})}^{\frac{1}{2}} - 2 \cdot 64^{\frac{1}{2}})$

Step 8.3.14

Multiply the exponents in ${(2^{3})}^{\frac{1}{2}}$ .

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Step 8.3.14.1

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

$\ln (8) \cdot 2^{\frac{5}{2}} - 16 \ln (64) - 2 (2 \cdot 2^{3 (\frac{1}{2})} - 2 \cdot 64^{\frac{1}{2}})$

Step 8.3.14.2

Combine $3$ and $\frac{1}{2}$ .

$\ln (8) \cdot 2^{\frac{5}{2}} - 16 \ln (64) - 2 (2 \cdot 2^{\frac{3}{2}} - 2 \cdot 64^{\frac{1}{2}})$

Step 8.3.15

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

$\ln (8) \cdot 2^{\frac{5}{2}} - 16 \ln (64) - 2 (2^{1 + \frac{3}{2}} - 2 \cdot 64^{\frac{1}{2}})$

Step 8.3.16

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

$\ln (8) \cdot 2^{\frac{5}{2}} - 16 \ln (64) - 2 (2^{\frac{2}{2} + \frac{3}{2}} - 2 \cdot 64^{\frac{1}{2}})$

Step 8.3.17

Combine the numerators over the common denominator.

$\ln (8) \cdot 2^{\frac{5}{2}} - 16 \ln (64) - 2 (2^{\frac{2 + 3}{2}} - 2 \cdot 64^{\frac{1}{2}})$

Step 8.3.18

Add $2$ and $3$ .

$\ln (8) \cdot 2^{\frac{5}{2}} - 16 \ln (64) - 2 (2^{\frac{5}{2}} - 2 \cdot 64^{\frac{1}{2}})$

Step 8.3.19

Rewrite $64$ as $8^{2}$ .

$\ln (8) \cdot 2^{\frac{5}{2}} - 16 \ln (64) - 2 (2^{\frac{5}{2}} - 2 \cdot {(8^{2})}^{\frac{1}{2}})$

Step 8.3.20

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

$\ln (8) \cdot 2^{\frac{5}{2}} - 16 \ln (64) - 2 (2^{\frac{5}{2}} - 2 \cdot 8^{2 (\frac{1}{2})})$

Step 8.3.21

Cancel the common factor of $2$ .

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Step 8.3.21.1

Cancel the common factor.

$\ln (8) \cdot 2^{\frac{5}{2}} - 16 \ln (64) - 2 (2^{\frac{5}{2}} - 2 \cdot 8^{2 (\frac{1}{2})})$

Step 8.3.21.2

Rewrite the expression.

$\ln (8) \cdot 2^{\frac{5}{2}} - 16 \ln (64) - 2 (2^{\frac{5}{2}} - 2 \cdot 8^{1})$

Step 8.3.22

Evaluate the exponent.

$\ln (8) \cdot 2^{\frac{5}{2}} - 16 \ln (64) - 2 (2^{\frac{5}{2}} - 2 \cdot 8)$

Step 8.3.23

Multiply $- 2$ by $8$ .

$\ln (8) \cdot 2^{\frac{5}{2}} - 16 \ln (64) - 2 (2^{\frac{5}{2}} - 16)$

Step 9

Simplify each term.

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Step 9.1

Rewrite $\ln (64)$ as $\ln (2^{6})$ .

$\ln (8) \cdot 2^{\frac{5}{2}} - 16 \ln (2^{6}) - 2 (2^{\frac{5}{2}} - 16)$

Step 9.2

Expand $\ln (2^{6})$ by moving $6$ outside the logarithm.

$\ln (8) \cdot 2^{\frac{5}{2}} - 16 (6 \ln (2)) - 2 (2^{\frac{5}{2}} - 16)$

Step 9.3

Multiply $6$ by $- 16$ .

$\ln (8) \cdot 2^{\frac{5}{2}} - 96 \ln (2) - 2 (2^{\frac{5}{2}} - 16)$

Step 9.4

Apply the distributive property.

$\ln (8) \cdot 2^{\frac{5}{2}} - 96 \ln (2) - 2 \cdot 2^{\frac{5}{2}} - 2 \cdot - 16$

Step 9.5

Multiply $- 2 \cdot 2^{\frac{5}{2}}$ .

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Step 9.5.1

Factor out negative.

$\ln (8) \cdot 2^{\frac{5}{2}} - 96 \ln (2) - (2 \cdot 2^{\frac{5}{2}}) - 2 \cdot - 16$

Step 9.5.2

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

$\ln (8) \cdot 2^{\frac{5}{2}} - 96 \ln (2) - (2^{1} \cdot 2^{\frac{5}{2}}) - 2 \cdot - 16$

Step 9.5.3

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

$\ln (8) \cdot 2^{\frac{5}{2}} - 96 \ln (2) - 2^{1 + \frac{5}{2}} - 2 \cdot - 16$

Step 9.5.4

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

$\ln (8) \cdot 2^{\frac{5}{2}} - 96 \ln (2) - 2^{\frac{2}{2} + \frac{5}{2}} - 2 \cdot - 16$

Step 9.5.5

Combine the numerators over the common denominator.

$\ln (8) \cdot 2^{\frac{5}{2}} - 96 \ln (2) - 2^{\frac{2 + 5}{2}} - 2 \cdot - 16$

Step 9.5.6

Add $2$ and $5$ .

$\ln (8) \cdot 2^{\frac{5}{2}} - 96 \ln (2) - 2^{\frac{7}{2}} - 2 \cdot - 16$

Step 9.6

Multiply $- 2$ by $- 16$ .

$\ln (8) \cdot 2^{\frac{5}{2}} - 96 \ln (2) - 2^{\frac{7}{2}} + 32$

Step 10

The result can be shown in multiple forms.

Exact Form:

$\ln (8) \cdot 2^{\frac{5}{2}} - 96 \ln (2) - 2^{\frac{7}{2}} + 32$

Decimal Form:

$- 34.09274011 \dots$