Calculus Examples

$\int_{36}^{49} \frac{\ln (y)}{\sqrt{y}} d y$

Step 1

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

$\int_{36}^{49} \ln (y) \frac{1}{\sqrt{y}} d y$

Step 2

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

$\ln (y) (2 y^{\frac{1}{2}})]_{36}^{49} - \int_{36}^{49} 2 y^{\frac{1}{2}} \frac{1}{y} d y$

Step 3

Simplify.

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Step 3.1

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

$\ln (y) (2 y^{\frac{1}{2}})]_{36}^{49} - \int_{36}^{49} y^{\frac{1}{2}} \frac{2}{y} d y$

Step 3.2

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

$\ln (y) (2 y^{\frac{1}{2}})]_{36}^{49} - \int_{36}^{49} \frac{y^{\frac{1}{2}} \cdot 2}{y} d y$

Step 3.3

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

$\ln (y) (2 y^{\frac{1}{2}})]_{36}^{49} - \int_{36}^{49} \frac{2 \cdot y^{\frac{1}{2}}}{y} d y$

Step 3.4

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

$\ln (y) (2 y^{\frac{1}{2}})]_{36}^{49} - \int_{36}^{49} \frac{2}{y \cdot y^{- \frac{1}{2}}} d y$

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 $1$ .

$\ln (y) (2 y^{\frac{1}{2}})]_{36}^{49} - \int_{36}^{49} \frac{2}{y^{1} y^{- \frac{1}{2}}} d y$

Step 3.5.1.2

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

$\ln (y) (2 y^{\frac{1}{2}})]_{36}^{49} - \int_{36}^{49} \frac{2}{y^{1 - \frac{1}{2}}} d y$

Step 3.5.2

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

$\ln (y) (2 y^{\frac{1}{2}})]_{36}^{49} - \int_{36}^{49} \frac{2}{y^{\frac{2}{2} - \frac{1}{2}}} d y$

Step 3.5.3

Combine the numerators over the common denominator.

$\ln (y) (2 y^{\frac{1}{2}})]_{36}^{49} - \int_{36}^{49} \frac{2}{y^{\frac{2 - 1}{2}}} d y$

Step 3.5.4

Subtract $1$ from $2$ .

$\ln (y) (2 y^{\frac{1}{2}})]_{36}^{49} - \int_{36}^{49} \frac{2}{y^{\frac{1}{2}}} d y$

Step 4

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

$\ln (y) (2 y^{\frac{1}{2}})]_{36}^{49} - (2 \int_{36}^{49} \frac{1}{y^{\frac{1}{2}}} d y)$

Step 5

Simplify the expression.

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Step 5.1

Multiply $2$ by $- 1$ .

$\ln (y) (2 y^{\frac{1}{2}})]_{36}^{49} - 2 \int_{36}^{49} \frac{1}{y^{\frac{1}{2}}} d y$

Step 5.2

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

$\ln (y) (2 y^{\frac{1}{2}})]_{36}^{49} - 2 \int_{36}^{49} {(y^{\frac{1}{2}})}^{- 1} d y$

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, ${(a^{m})}^{n} = a^{m n}$ .

$\ln (y) (2 y^{\frac{1}{2}})]_{36}^{49} - 2 \int_{36}^{49} y^{\frac{1}{2} \cdot - 1} d y$

Step 5.3.2

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

$\ln (y) (2 y^{\frac{1}{2}})]_{36}^{49} - 2 \int_{36}^{49} y^{\frac{- 1}{2}} d y$

Step 5.3.3

Move the negative in front of the fraction.

$\ln (y) (2 y^{\frac{1}{2}})]_{36}^{49} - 2 \int_{36}^{49} y^{- \frac{1}{2}} d y$

Step 6

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

$\ln (y) (2 y^{\frac{1}{2}})]_{36}^{49} - 2 (2 y^{\frac{1}{2}}]_{36}^{49})$

Step 7

Substitute and simplify.

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Step 7.1

Evaluate $\ln (y) (2 y^{\frac{1}{2}})$ at $49$ and at $36$ .

$(\ln (49) (2 \cdot 49^{\frac{1}{2}})) - \ln (36) (2 \cdot 36^{\frac{1}{2}}) - 2 (2 y^{\frac{1}{2}}]_{36}^{49})$

Step 7.2

Evaluate $2 y^{\frac{1}{2}}$ at $49$ and at $36$ .

$(\ln (49) (2 \cdot 49^{\frac{1}{2}})) - \ln (36) (2 \cdot 36^{\frac{1}{2}}) - 2 ((2 \cdot 49^{\frac{1}{2}}) - 2 \cdot 36^{\frac{1}{2}})$

Step 7.3

Simplify.

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Step 7.3.1

Rewrite $49$ as $7^{2}$ .

$\ln (49) (2 \cdot {(7^{2})}^{\frac{1}{2}}) - \ln (36) (2 \cdot 36^{\frac{1}{2}}) - 2 ((2 \cdot 49^{\frac{1}{2}}) - 2 \cdot 36^{\frac{1}{2}})$

Step 7.3.2

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

$\ln (49) (2 \cdot 7^{2 (\frac{1}{2})}) - \ln (36) (2 \cdot 36^{\frac{1}{2}}) - 2 ((2 \cdot 49^{\frac{1}{2}}) - 2 \cdot 36^{\frac{1}{2}})$

Step 7.3.3

Cancel the common factor of $2$ .

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Step 7.3.3.1

Cancel the common factor.

$\ln (49) (2 \cdot 7^{2 (\frac{1}{2})}) - \ln (36) (2 \cdot 36^{\frac{1}{2}}) - 2 ((2 \cdot 49^{\frac{1}{2}}) - 2 \cdot 36^{\frac{1}{2}})$

Step 7.3.3.2

Rewrite the expression.

$\ln (49) (2 \cdot 7^{1}) - \ln (36) (2 \cdot 36^{\frac{1}{2}}) - 2 ((2 \cdot 49^{\frac{1}{2}}) - 2 \cdot 36^{\frac{1}{2}})$

Step 7.3.4

Evaluate the exponent.

$\ln (49) (2 \cdot 7) - \ln (36) (2 \cdot 36^{\frac{1}{2}}) - 2 ((2 \cdot 49^{\frac{1}{2}}) - 2 \cdot 36^{\frac{1}{2}})$

Step 7.3.5

Multiply $2$ by $7$ .

$\ln (49) \cdot 14 - \ln (36) (2 \cdot 36^{\frac{1}{2}}) - 2 ((2 \cdot 49^{\frac{1}{2}}) - 2 \cdot 36^{\frac{1}{2}})$

Step 7.3.6

Move $14$ to the left of $\ln (49)$ .

$14 \cdot \ln (49) - \ln (36) (2 \cdot 36^{\frac{1}{2}}) - 2 ((2 \cdot 49^{\frac{1}{2}}) - 2 \cdot 36^{\frac{1}{2}})$

Step 7.3.7

Rewrite $36$ as $6^{2}$ .

$14 \ln (49) - \ln (36) (2 \cdot {(6^{2})}^{\frac{1}{2}}) - 2 ((2 \cdot 49^{\frac{1}{2}}) - 2 \cdot 36^{\frac{1}{2}})$

Step 7.3.8

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

$14 \ln (49) - \ln (36) (2 \cdot 6^{2 (\frac{1}{2})}) - 2 ((2 \cdot 49^{\frac{1}{2}}) - 2 \cdot 36^{\frac{1}{2}})$

Step 7.3.9

Cancel the common factor of $2$ .

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Step 7.3.9.1

Cancel the common factor.

$14 \ln (49) - \ln (36) (2 \cdot 6^{2 (\frac{1}{2})}) - 2 ((2 \cdot 49^{\frac{1}{2}}) - 2 \cdot 36^{\frac{1}{2}})$

Step 7.3.9.2

Rewrite the expression.

$14 \ln (49) - \ln (36) (2 \cdot 6^{1}) - 2 ((2 \cdot 49^{\frac{1}{2}}) - 2 \cdot 36^{\frac{1}{2}})$

Step 7.3.10

Evaluate the exponent.

$14 \ln (49) - \ln (36) (2 \cdot 6) - 2 ((2 \cdot 49^{\frac{1}{2}}) - 2 \cdot 36^{\frac{1}{2}})$

Step 7.3.11

Multiply $2$ by $6$ .

$14 \ln (49) - \ln (36) \cdot 12 - 2 ((2 \cdot 49^{\frac{1}{2}}) - 2 \cdot 36^{\frac{1}{2}})$

Step 7.3.12

Multiply $12$ by $- 1$ .

$14 \ln (49) - 12 \ln (36) - 2 ((2 \cdot 49^{\frac{1}{2}}) - 2 \cdot 36^{\frac{1}{2}})$

Step 7.3.13

Rewrite $49$ as $7^{2}$ .

$14 \ln (49) - 12 \ln (36) - 2 (2 \cdot {(7^{2})}^{\frac{1}{2}} - 2 \cdot 36^{\frac{1}{2}})$

Step 7.3.14

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

$14 \ln (49) - 12 \ln (36) - 2 (2 \cdot 7^{2 (\frac{1}{2})} - 2 \cdot 36^{\frac{1}{2}})$

Step 7.3.15

Cancel the common factor of $2$ .

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Step 7.3.15.1

Cancel the common factor.

$14 \ln (49) - 12 \ln (36) - 2 (2 \cdot 7^{2 (\frac{1}{2})} - 2 \cdot 36^{\frac{1}{2}})$

Step 7.3.15.2

Rewrite the expression.

$14 \ln (49) - 12 \ln (36) - 2 (2 \cdot 7^{1} - 2 \cdot 36^{\frac{1}{2}})$

Step 7.3.16

Evaluate the exponent.

$14 \ln (49) - 12 \ln (36) - 2 (2 \cdot 7 - 2 \cdot 36^{\frac{1}{2}})$

Step 7.3.17

Multiply $2$ by $7$ .

$14 \ln (49) - 12 \ln (36) - 2 (14 - 2 \cdot 36^{\frac{1}{2}})$

Step 7.3.18

Rewrite $36$ as $6^{2}$ .

$14 \ln (49) - 12 \ln (36) - 2 (14 - 2 \cdot {(6^{2})}^{\frac{1}{2}})$

Step 7.3.19

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

$14 \ln (49) - 12 \ln (36) - 2 (14 - 2 \cdot 6^{2 (\frac{1}{2})})$

Step 7.3.20

Cancel the common factor of $2$ .

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Step 7.3.20.1

Cancel the common factor.

$14 \ln (49) - 12 \ln (36) - 2 (14 - 2 \cdot 6^{2 (\frac{1}{2})})$

Step 7.3.20.2

Rewrite the expression.

$14 \ln (49) - 12 \ln (36) - 2 (14 - 2 \cdot 6^{1})$

Step 7.3.21

Evaluate the exponent.

$14 \ln (49) - 12 \ln (36) - 2 (14 - 2 \cdot 6)$

Step 7.3.22

Multiply $- 2$ by $6$ .

$14 \ln (49) - 12 \ln (36) - 2 (14 - 12)$

Step 7.3.23

Subtract $12$ from $14$ .

$14 \ln (49) - 12 \ln (36) - 2 \cdot 2$

Step 7.3.24

Multiply $- 2$ by $2$ .

$14 \ln (49) - 12 \ln (36) - 4$

Step 8

Simplify each term.

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Step 8.1

Rewrite $\ln (49)$ as $\ln (7^{2})$ .

$14 \ln (7^{2}) - 12 \ln (36) - 4$

Step 8.2

Expand $\ln (7^{2})$ by moving $2$ outside the logarithm.

$14 (2 \ln (7)) - 12 \ln (36) - 4$

Step 8.3

Multiply $2$ by $14$ .

$28 \ln (7) - 12 \ln (36) - 4$

Step 9

The result can be shown in multiple forms.

Exact Form:

$28 \ln (7) - 12 \ln (36) - 4$

Decimal Form:

$7.48325691 \dots$