Calculus Examples

$\int_{4}^{9} \frac{\ln (y)}{\sqrt{y}} d y$

Step 1

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

$\int_{4}^{9} \ln (y) \frac{1}{\sqrt{y}} d y$

Step 2

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

$\ln (y) (2 y^{\frac{1}{2}})]_{4}^{9} - \int_{4}^{9} 2 y^{\frac{1}{2}} \frac{1}{y} d y$

Step 3

Simplify.

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Step 3.1

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

$\ln (y) (2 y^{\frac{1}{2}})]_{4}^{9} - \int_{4}^{9} y^{\frac{1}{2}} \frac{2}{y} d y$

Step 3.2

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

$\ln (y) (2 y^{\frac{1}{2}})]_{4}^{9} - \int_{4}^{9} \frac{y^{\frac{1}{2}} \cdot 2}{y} d y$

Step 3.3

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

$\ln (y) (2 y^{\frac{1}{2}})]_{4}^{9} - \int_{4}^{9} \frac{2 \cdot y^{\frac{1}{2}}}{y} d y$

Step 3.4

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

$\ln (y) (2 y^{\frac{1}{2}})]_{4}^{9} - \int_{4}^{9} \frac{2}{y \cdot y^{- \frac{1}{2}}} d y$

Step 3.5

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

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Step 3.5.1

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

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Step 3.5.1.1

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

$\ln (y) (2 y^{\frac{1}{2}})]_{4}^{9} - \int_{4}^{9} \frac{2}{y^{1} y^{- \frac{1}{2}}} d y$

Step 3.5.1.2

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

$\ln (y) (2 y^{\frac{1}{2}})]_{4}^{9} - \int_{4}^{9} \frac{2}{y^{1 - \frac{1}{2}}} d y$

Step 3.5.2

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

$\ln (y) (2 y^{\frac{1}{2}})]_{4}^{9} - \int_{4}^{9} \frac{2}{y^{\frac{2}{2} - \frac{1}{2}}} d y$

Step 3.5.3

Combine the numerators over the common denominator.

$\ln (y) (2 y^{\frac{1}{2}})]_{4}^{9} - \int_{4}^{9} \frac{2}{y^{\frac{2 - 1}{2}}} d y$

Step 3.5.4

Subtract $1$ from $2$ .

$\ln (y) (2 y^{\frac{1}{2}})]_{4}^{9} - \int_{4}^{9} \frac{2}{y^{\frac{1}{2}}} d y$

Step 4

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

$\ln (y) (2 y^{\frac{1}{2}})]_{4}^{9} - (2 \int_{4}^{9} \frac{1}{y^{\frac{1}{2}}} d y)$

Step 5

Simplify the expression.

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Step 5.1

Multiply $2$ by $- 1$ .

$\ln (y) (2 y^{\frac{1}{2}})]_{4}^{9} - 2 \int_{4}^{9} \frac{1}{y^{\frac{1}{2}}} d y$

Step 5.2

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

$\ln (y) (2 y^{\frac{1}{2}})]_{4}^{9} - 2 \int_{4}^{9} {(y^{\frac{1}{2}})}^{- 1} d y$

Step 5.3

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

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Step 5.3.1

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

$\ln (y) (2 y^{\frac{1}{2}})]_{4}^{9} - 2 \int_{4}^{9} y^{\frac{1}{2} \cdot - 1} d y$

Step 5.3.2

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

$\ln (y) (2 y^{\frac{1}{2}})]_{4}^{9} - 2 \int_{4}^{9} y^{\frac{- 1}{2}} d y$

Step 5.3.3

Move the negative in front of the fraction.

$\ln (y) (2 y^{\frac{1}{2}})]_{4}^{9} - 2 \int_{4}^{9} y^{- \frac{1}{2}} d y$

Step 6

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

$\ln (y) (2 y^{\frac{1}{2}})]_{4}^{9} - 2 (2 y^{\frac{1}{2}}]_{4}^{9})$

Step 7

Substitute and simplify.

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Step 7.1

Evaluate $\ln (y) (2 y^{\frac{1}{2}})$ at $9$ and at $4$ .

$(\ln (9) (2 \cdot 9^{\frac{1}{2}})) - \ln (4) (2 \cdot 4^{\frac{1}{2}}) - 2 (2 y^{\frac{1}{2}}]_{4}^{9})$

Step 7.2

Evaluate $2 y^{\frac{1}{2}}$ at $9$ and at $4$ .

$(\ln (9) (2 \cdot 9^{\frac{1}{2}})) - \ln (4) (2 \cdot 4^{\frac{1}{2}}) - 2 ((2 \cdot 9^{\frac{1}{2}}) - 2 \cdot 4^{\frac{1}{2}})$

Step 7.3

Simplify.

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Step 7.3.1

Rewrite $9$ as $3^{2}$ .

$\ln (9) (2 \cdot {(3^{2})}^{\frac{1}{2}}) - \ln (4) (2 \cdot 4^{\frac{1}{2}}) - 2 ((2 \cdot 9^{\frac{1}{2}}) - 2 \cdot 4^{\frac{1}{2}})$

Step 7.3.2

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

$\ln (9) (2 \cdot 3^{2 (\frac{1}{2})}) - \ln (4) (2 \cdot 4^{\frac{1}{2}}) - 2 ((2 \cdot 9^{\frac{1}{2}}) - 2 \cdot 4^{\frac{1}{2}})$

Step 7.3.3

Cancel the common factor of $2$ .

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Step 7.3.3.1

Cancel the common factor.

$\ln (9) (2 \cdot 3^{2 (\frac{1}{2})}) - \ln (4) (2 \cdot 4^{\frac{1}{2}}) - 2 ((2 \cdot 9^{\frac{1}{2}}) - 2 \cdot 4^{\frac{1}{2}})$

Step 7.3.3.2

Rewrite the expression.

$\ln (9) (2 \cdot 3^{1}) - \ln (4) (2 \cdot 4^{\frac{1}{2}}) - 2 ((2 \cdot 9^{\frac{1}{2}}) - 2 \cdot 4^{\frac{1}{2}})$

Step 7.3.4

Evaluate the exponent.

$\ln (9) (2 \cdot 3) - \ln (4) (2 \cdot 4^{\frac{1}{2}}) - 2 ((2 \cdot 9^{\frac{1}{2}}) - 2 \cdot 4^{\frac{1}{2}})$

Step 7.3.5

Multiply $2$ by $3$ .

$\ln (9) \cdot 6 - \ln (4) (2 \cdot 4^{\frac{1}{2}}) - 2 ((2 \cdot 9^{\frac{1}{2}}) - 2 \cdot 4^{\frac{1}{2}})$

Step 7.3.6

Move $6$ to the left of $\ln (9)$ .

$6 \cdot \ln (9) - \ln (4) (2 \cdot 4^{\frac{1}{2}}) - 2 ((2 \cdot 9^{\frac{1}{2}}) - 2 \cdot 4^{\frac{1}{2}})$

Step 7.3.7

Rewrite $4$ as $2^{2}$ .

$6 \ln (9) - \ln (4) (2 \cdot {(2^{2})}^{\frac{1}{2}}) - 2 ((2 \cdot 9^{\frac{1}{2}}) - 2 \cdot 4^{\frac{1}{2}})$

Step 7.3.8

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

$6 \ln (9) - \ln (4) (2 \cdot 2^{2 (\frac{1}{2})}) - 2 ((2 \cdot 9^{\frac{1}{2}}) - 2 \cdot 4^{\frac{1}{2}})$

Step 7.3.9

Cancel the common factor of $2$ .

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Step 7.3.9.1

Cancel the common factor.

$6 \ln (9) - \ln (4) (2 \cdot 2^{2 (\frac{1}{2})}) - 2 ((2 \cdot 9^{\frac{1}{2}}) - 2 \cdot 4^{\frac{1}{2}})$

Step 7.3.9.2

Rewrite the expression.

$6 \ln (9) - \ln (4) (2 \cdot 2^{1}) - 2 ((2 \cdot 9^{\frac{1}{2}}) - 2 \cdot 4^{\frac{1}{2}})$

Step 7.3.10

Evaluate the exponent.

$6 \ln (9) - \ln (4) (2 \cdot 2) - 2 ((2 \cdot 9^{\frac{1}{2}}) - 2 \cdot 4^{\frac{1}{2}})$

Step 7.3.11

Multiply $2$ by $2$ .

$6 \ln (9) - \ln (4) \cdot 4 - 2 ((2 \cdot 9^{\frac{1}{2}}) - 2 \cdot 4^{\frac{1}{2}})$

Step 7.3.12

Multiply $4$ by $- 1$ .

$6 \ln (9) - 4 \ln (4) - 2 ((2 \cdot 9^{\frac{1}{2}}) - 2 \cdot 4^{\frac{1}{2}})$

Step 7.3.13

Rewrite $9$ as $3^{2}$ .

$6 \ln (9) - 4 \ln (4) - 2 (2 \cdot {(3^{2})}^{\frac{1}{2}} - 2 \cdot 4^{\frac{1}{2}})$

Step 7.3.14

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

$6 \ln (9) - 4 \ln (4) - 2 (2 \cdot 3^{2 (\frac{1}{2})} - 2 \cdot 4^{\frac{1}{2}})$

Step 7.3.15

Cancel the common factor of $2$ .

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Step 7.3.15.1

Cancel the common factor.

$6 \ln (9) - 4 \ln (4) - 2 (2 \cdot 3^{2 (\frac{1}{2})} - 2 \cdot 4^{\frac{1}{2}})$

Step 7.3.15.2

Rewrite the expression.

$6 \ln (9) - 4 \ln (4) - 2 (2 \cdot 3^{1} - 2 \cdot 4^{\frac{1}{2}})$

Step 7.3.16

Evaluate the exponent.

$6 \ln (9) - 4 \ln (4) - 2 (2 \cdot 3 - 2 \cdot 4^{\frac{1}{2}})$

Step 7.3.17

Multiply $2$ by $3$ .

$6 \ln (9) - 4 \ln (4) - 2 (6 - 2 \cdot 4^{\frac{1}{2}})$

Step 7.3.18

Rewrite $4$ as $2^{2}$ .

$6 \ln (9) - 4 \ln (4) - 2 (6 - 2 \cdot {(2^{2})}^{\frac{1}{2}})$

Step 7.3.19

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

$6 \ln (9) - 4 \ln (4) - 2 (6 - 2 \cdot 2^{2 (\frac{1}{2})})$

Step 7.3.20

Cancel the common factor of $2$ .

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Step 7.3.20.1

Cancel the common factor.

$6 \ln (9) - 4 \ln (4) - 2 (6 - 2 \cdot 2^{2 (\frac{1}{2})})$

Step 7.3.20.2

Rewrite the expression.

$6 \ln (9) - 4 \ln (4) - 2 (6 - 2 \cdot 2^{1})$

Step 7.3.21

Evaluate the exponent.

$6 \ln (9) - 4 \ln (4) - 2 (6 - 2 \cdot 2)$

Step 7.3.22

Multiply $- 2$ by $2$ .

$6 \ln (9) - 4 \ln (4) - 2 (6 - 4)$

Step 7.3.23

Subtract $4$ from $6$ .

$6 \ln (9) - 4 \ln (4) - 2 \cdot 2$

Step 7.3.24

Multiply $- 2$ by $2$ .

$6 \ln (9) - 4 \ln (4) - 4$

Step 8

Simplify each term.

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Step 8.1

Rewrite $\ln (4)$ as $\ln (2^{2})$ .

$6 \ln (9) - 4 \ln (2^{2}) - 4$

Step 8.2

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

$6 \ln (9) - 4 (2 \ln (2)) - 4$

Step 8.3

Multiply $2$ by $- 4$ .

$6 \ln (9) - 8 \ln (2) - 4$

Step 9

The result can be shown in multiple forms.

Exact Form:

$6 \ln (9) - 8 \ln (2) - 4$

Decimal Form:

$3.63817001 \dots$