Calculus Examples

$\int_{1}^{3} \frac{\ln^{2} (x)}{x^{3}} d x$

Step 1

Apply basic rules of exponents.

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Step 1.1

Move $x^{3}$ out of the denominator by raising it to the $- 1$ power.

$\int_{1}^{3} \ln^{2} (x) {(x^{3})}^{- 1} d x$

Step 1.2

Multiply the exponents in ${(x^{3})}^{- 1}$ .

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Step 1.2.1

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

$\int_{1}^{3} \ln^{2} (x) x^{3 \cdot - 1} d x$

Step 1.2.2

Multiply $3$ by $- 1$ .

$\int_{1}^{3} \ln^{2} (x) x^{- 3} d x$

Step 2

Integrate by parts using the formula $\int u d v = u v - \int v d u$ , where $u = \ln^{2} (x)$ and $d v = x^{- 3}$ .

$\ln^{2} (x) (- \frac{1}{2 x^{2}})]_{1}^{3} - \int_{1}^{3} - \frac{1}{2 x^{2}} \cdot \frac{\ln (x^{2})}{x} d x$

Step 3

Simplify.

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Step 3.1

Combine $\ln^{2} (x)$ and $\frac{1}{2 x^{2}}$ .

$- \frac{\ln^{2} (x)}{2 x^{2}}]_{1}^{3} - \int_{1}^{3} - \frac{1}{2 x^{2}} \cdot \frac{\ln (x^{2})}{x} d x$

Step 3.2

Multiply $\frac{\ln (x^{2})}{x}$ by $\frac{1}{2 x^{2}}$ .

$- \frac{\ln^{2} (x)}{2 x^{2}}]_{1}^{3} - \int_{1}^{3} - \frac{\ln (x^{2})}{x (2 x^{2})} d x$

Step 3.3

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

$- \frac{\ln^{2} (x)}{2 x^{2}}]_{1}^{3} - \int_{1}^{3} - \frac{\ln (x^{2})}{2 (x^{1} x^{2})} d x$

Step 3.4

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

$- \frac{\ln^{2} (x)}{2 x^{2}}]_{1}^{3} - \int_{1}^{3} - \frac{\ln (x^{2})}{2 x^{1 + 2}} d x$

Step 3.5

Add $1$ and $2$ .

$- \frac{\ln^{2} (x)}{2 x^{2}}]_{1}^{3} - \int_{1}^{3} - \frac{\ln (x^{2})}{2 x^{3}} d x$

Step 4

Rewrite $- \frac{\ln (x^{2})}{2 x^{3}}$ as $- \frac{2 \ln (x)}{2 x^{3}}$ .

$- \frac{\ln^{2} (x)}{2 x^{2}}]_{1}^{3} - \int_{1}^{3} - \frac{2 \ln (x)}{2 x^{3}} d x$

Step 5

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

$- \frac{\ln^{2} (x)}{2 x^{2}}]_{1}^{3} - - \int_{1}^{3} \frac{2 \ln (x)}{2 x^{3}} d x$

Step 6

Simplify the expression.

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Step 6.1

Simplify.

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Step 6.1.1

Cancel the common factor of $2$ .

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Step 6.1.1.1

Cancel the common factor.

$- \frac{\ln^{2} (x)}{2 x^{2}}]_{1}^{3} - - \int_{1}^{3} \frac{2 \ln (x)}{2 x^{3}} d x$

Step 6.1.1.2

Rewrite the expression.

$- \frac{\ln^{2} (x)}{2 x^{2}}]_{1}^{3} - - \int_{1}^{3} \frac{\ln (x)}{x^{3}} d x$

Step 6.1.2

Multiply $- 1$ by $- 1$ .

$- \frac{\ln^{2} (x)}{2 x^{2}}]_{1}^{3} + 1 \int_{1}^{3} \frac{\ln (x)}{x^{3}} d x$

Step 6.1.3

Multiply $\int_{1}^{3} \frac{\ln (x)}{x^{3}} d x$ by $1$ .

$- \frac{\ln^{2} (x)}{2 x^{2}}]_{1}^{3} + \int_{1}^{3} \frac{\ln (x)}{x^{3}} d x$

Step 6.2

Apply basic rules of exponents.

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Step 6.2.1

Move $x^{3}$ out of the denominator by raising it to the $- 1$ power.

$- \frac{\ln^{2} (x)}{2 x^{2}}]_{1}^{3} + \int_{1}^{3} \ln (x) {(x^{3})}^{- 1} d x$

Step 6.2.2

Multiply the exponents in ${(x^{3})}^{- 1}$ .

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Step 6.2.2.1

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

$- \frac{\ln^{2} (x)}{2 x^{2}}]_{1}^{3} + \int_{1}^{3} \ln (x) x^{3 \cdot - 1} d x$

Step 6.2.2.2

Multiply $3$ by $- 1$ .

$- \frac{\ln^{2} (x)}{2 x^{2}}]_{1}^{3} + \int_{1}^{3} \ln (x) x^{- 3} d x$

Step 7

Integrate by parts using the formula $\int u d v = u v - \int v d u$ , where $u = \ln (x)$ and $d v = x^{- 3}$ .

$- \frac{\ln^{2} (x)}{2 x^{2}}]_{1}^{3} + \ln (x) (- \frac{1}{2 x^{2}})]_{1}^{3} - \int_{1}^{3} - \frac{1}{2 x^{2}} \cdot \frac{1}{x} d x$

Step 8

Simplify.

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Step 8.1

Combine $\ln (x)$ and $\frac{1}{2 x^{2}}$ .

$- \frac{\ln^{2} (x)}{2 x^{2}}]_{1}^{3} + - \frac{\ln (x)}{2 x^{2}}]_{1}^{3} - \int_{1}^{3} - \frac{1}{2 x^{2}} \cdot \frac{1}{x} d x$

Step 8.2

Multiply $\frac{1}{x}$ by $\frac{1}{2 x^{2}}$ .

$- \frac{\ln^{2} (x)}{2 x^{2}}]_{1}^{3} + - \frac{\ln (x)}{2 x^{2}}]_{1}^{3} - \int_{1}^{3} - \frac{1}{x (2 x^{2})} d x$

Step 8.3

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

$- \frac{\ln^{2} (x)}{2 x^{2}}]_{1}^{3} + - \frac{\ln (x)}{2 x^{2}}]_{1}^{3} - \int_{1}^{3} - \frac{1}{2 (x^{1} x^{2})} d x$

Step 8.4

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

$- \frac{\ln^{2} (x)}{2 x^{2}}]_{1}^{3} + - \frac{\ln (x)}{2 x^{2}}]_{1}^{3} - \int_{1}^{3} - \frac{1}{2 x^{1 + 2}} d x$

Step 8.5

Add $1$ and $2$ .

$- \frac{\ln^{2} (x)}{2 x^{2}}]_{1}^{3} + - \frac{\ln (x)}{2 x^{2}}]_{1}^{3} - \int_{1}^{3} - \frac{1}{2 x^{3}} d x$

Step 9

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

$- \frac{\ln^{2} (x)}{2 x^{2}}]_{1}^{3} + - \frac{\ln (x)}{2 x^{2}}]_{1}^{3} - - \int_{1}^{3} \frac{1}{2 x^{3}} d x$

Step 10

Simplify.

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Step 10.1

Multiply $- 1$ by $- 1$ .

$- \frac{\ln^{2} (x)}{2 x^{2}}]_{1}^{3} + - \frac{\ln (x)}{2 x^{2}}]_{1}^{3} + 1 \int_{1}^{3} \frac{1}{2 x^{3}} d x$

Step 10.2

Multiply $\int_{1}^{3} \frac{1}{2 x^{3}} d x$ by $1$ .

$- \frac{\ln^{2} (x)}{2 x^{2}}]_{1}^{3} + - \frac{\ln (x)}{2 x^{2}}]_{1}^{3} + \int_{1}^{3} \frac{1}{2 x^{3}} d x$

Step 11

Since $\frac{1}{2}$ is constant with respect to $x$ , move $\frac{1}{2}$ out of the integral.

$- \frac{\ln^{2} (x)}{2 x^{2}}]_{1}^{3} + - \frac{\ln (x)}{2 x^{2}}]_{1}^{3} + \frac{1}{2} \int_{1}^{3} \frac{1}{x^{3}} d x$

Step 12

Apply basic rules of exponents.

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Step 12.1

Move $x^{3}$ out of the denominator by raising it to the $- 1$ power.

$- \frac{\ln^{2} (x)}{2 x^{2}}]_{1}^{3} + - \frac{\ln (x)}{2 x^{2}}]_{1}^{3} + \frac{1}{2} \int_{1}^{3} {(x^{3})}^{- 1} d x$

Step 12.2

Multiply the exponents in ${(x^{3})}^{- 1}$ .

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Step 12.2.1

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

$- \frac{\ln^{2} (x)}{2 x^{2}}]_{1}^{3} + - \frac{\ln (x)}{2 x^{2}}]_{1}^{3} + \frac{1}{2} \int_{1}^{3} x^{3 \cdot - 1} d x$

Step 12.2.2

Multiply $3$ by $- 1$ .

$- \frac{\ln^{2} (x)}{2 x^{2}}]_{1}^{3} + - \frac{\ln (x)}{2 x^{2}}]_{1}^{3} + \frac{1}{2} \int_{1}^{3} x^{- 3} d x$

Step 13

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

$- \frac{\ln^{2} (x)}{2 x^{2}}]_{1}^{3} + - \frac{\ln (x)}{2 x^{2}}]_{1}^{3} + \frac{1}{2} - \frac{1}{2} x^{- 2}]_{1}^{3}$

Step 14

Simplify the answer.

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Step 14.1

Combine $- \frac{\ln^{2} (x)}{2 x^{2}}]_{1}^{3}$ and $- \frac{\ln (x)}{2 x^{2}}]_{1}^{3}$ .

$- \frac{\ln^{2} (x)}{2 x^{2}} - \frac{\ln (x)}{2 x^{2}}]_{1}^{3} + \frac{1}{2} - \frac{1}{2} x^{- 2}]_{1}^{3}$

Step 14.2

Substitute and simplify.

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Step 14.2.1

Evaluate $- \frac{\ln^{2} (x)}{2 x^{2}} - \frac{\ln (x)}{2 x^{2}}$ at $3$ and at $1$ .

$(- \frac{\ln^{2} (3)}{2 \cdot 3^{2}} - \frac{\ln (3)}{2 \cdot 3^{2}}) - (- \frac{\ln^{2} (1)}{2 \cdot 1^{2}} - \frac{\ln (1)}{2 \cdot 1^{2}}) + \frac{1}{2} - \frac{1}{2} x^{- 2}]_{1}^{3}$

Step 14.2.2

Evaluate $- \frac{1}{2} x^{- 2}$ at $3$ and at $1$ .

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

Step 14.2.3

Simplify.

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Step 14.2.3.1

Raise $3$ to the power of $2$ .

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

Step 14.2.3.2

Multiply $2$ by $9$ .

$- \frac{\ln^{2} (3)}{18} - \frac{\ln (3)}{2 \cdot 3^{2}} - (- \frac{\ln^{2} (1)}{2 \cdot 1^{2}} - \frac{\ln (1)}{2 \cdot 1^{2}}) + \frac{1}{2} ((- \frac{1}{2} \cdot 3^{- 2}) + \frac{1}{2} \cdot 1^{- 2})$

Step 14.2.3.3

Raise $3$ to the power of $2$ .

$- \frac{\ln^{2} (3)}{18} - \frac{\ln (3)}{2 \cdot 9} - (- \frac{\ln^{2} (1)}{2 \cdot 1^{2}} - \frac{\ln (1)}{2 \cdot 1^{2}}) + \frac{1}{2} ((- \frac{1}{2} \cdot 3^{- 2}) + \frac{1}{2} \cdot 1^{- 2})$

Step 14.2.3.4

Multiply $2$ by $9$ .

$- \frac{\ln^{2} (3)}{18} - \frac{\ln (3)}{18} - (- \frac{\ln^{2} (1)}{2 \cdot 1^{2}} - \frac{\ln (1)}{2 \cdot 1^{2}}) + \frac{1}{2} ((- \frac{1}{2} \cdot 3^{- 2}) + \frac{1}{2} \cdot 1^{- 2})$

Step 14.2.3.5

One to any power is one.

$- \frac{\ln^{2} (3)}{18} - \frac{\ln (3)}{18} - (- \frac{\ln^{2} (1)}{2 \cdot 1} - \frac{\ln (1)}{2 \cdot 1^{2}}) + \frac{1}{2} ((- \frac{1}{2} \cdot 3^{- 2}) + \frac{1}{2} \cdot 1^{- 2})$

Step 14.2.3.6

Multiply $2$ by $1$ .

$- \frac{\ln^{2} (3)}{18} - \frac{\ln (3)}{18} - (- \frac{\ln^{2} (1)}{2} - \frac{\ln (1)}{2 \cdot 1^{2}}) + \frac{1}{2} ((- \frac{1}{2} \cdot 3^{- 2}) + \frac{1}{2} \cdot 1^{- 2})$

Step 14.2.3.7

One to any power is one.

$- \frac{\ln^{2} (3)}{18} - \frac{\ln (3)}{18} - (- \frac{\ln^{2} (1)}{2} - \frac{\ln (1)}{2 \cdot 1}) + \frac{1}{2} ((- \frac{1}{2} \cdot 3^{- 2}) + \frac{1}{2} \cdot 1^{- 2})$

Step 14.2.3.8

Multiply $2$ by $1$ .

$- \frac{\ln^{2} (3)}{18} - \frac{\ln (3)}{18} - (- \frac{\ln^{2} (1)}{2} - \frac{\ln (1)}{2}) + \frac{1}{2} ((- \frac{1}{2} \cdot 3^{- 2}) + \frac{1}{2} \cdot 1^{- 2})$

Step 14.2.3.9

To write $- (- \frac{\ln^{2} (1)}{2} - \frac{\ln (1)}{2})$ as a fraction with a common denominator, multiply by $\frac{18}{18}$ .

$- \frac{\ln^{2} (3)}{18} - (- \frac{\ln^{2} (1)}{2} - \frac{\ln (1)}{2}) \cdot \frac{18}{18} - \frac{\ln (3)}{18} + \frac{1}{2} ((- \frac{1}{2} \cdot 3^{- 2}) + \frac{1}{2} \cdot 1^{- 2})$

Step 14.2.3.10

Combine $- (- \frac{\ln^{2} (1)}{2} - \frac{\ln (1)}{2})$ and $\frac{18}{18}$ .

$- \frac{\ln^{2} (3)}{18} + \frac{- (- \frac{\ln^{2} (1)}{2} - \frac{\ln (1)}{2}) \cdot 18}{18} - \frac{\ln (3)}{18} + \frac{1}{2} ((- \frac{1}{2} \cdot 3^{- 2}) + \frac{1}{2} \cdot 1^{- 2})$

Step 14.2.3.11

Combine the numerators over the common denominator.

$\frac{- \ln^{2} (3) - (- \frac{\ln^{2} (1)}{2} - \frac{\ln (1)}{2}) \cdot 18}{18} - \frac{\ln (3)}{18} + \frac{1}{2} ((- \frac{1}{2} \cdot 3^{- 2}) + \frac{1}{2} \cdot 1^{- 2})$

Step 14.2.3.12

Multiply $18$ by $- 1$ .

$\frac{- \ln^{2} (3) - 18 (- \frac{\ln^{2} (1)}{2} - \frac{\ln (1)}{2})}{18} - \frac{\ln (3)}{18} + \frac{1}{2} ((- \frac{1}{2} \cdot 3^{- 2}) + \frac{1}{2} \cdot 1^{- 2})$

Step 14.2.3.13

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

$\frac{- \ln^{2} (3) - 18 (- \frac{\ln^{2} (1)}{2} - \frac{\ln (1)}{2})}{18} - \frac{\ln (3)}{18} + \frac{1}{2} (- \frac{1}{2} \cdot \frac{1}{3^{2}} + \frac{1}{2} \cdot 1^{- 2})$

Step 14.2.3.14

Raise $3$ to the power of $2$ .

$\frac{- \ln^{2} (3) - 18 (- \frac{\ln^{2} (1)}{2} - \frac{\ln (1)}{2})}{18} - \frac{\ln (3)}{18} + \frac{1}{2} (- \frac{1}{2} \cdot \frac{1}{9} + \frac{1}{2} \cdot 1^{- 2})$

Step 14.2.3.15

Multiply $\frac{1}{9}$ by $\frac{1}{2}$ .

$\frac{- \ln^{2} (3) - 18 (- \frac{\ln^{2} (1)}{2} - \frac{\ln (1)}{2})}{18} - \frac{\ln (3)}{18} + \frac{1}{2} (- \frac{1}{9 \cdot 2} + \frac{1}{2} \cdot 1^{- 2})$

Step 14.2.3.16

Multiply $9$ by $2$ .

$\frac{- \ln^{2} (3) - 18 (- \frac{\ln^{2} (1)}{2} - \frac{\ln (1)}{2})}{18} - \frac{\ln (3)}{18} + \frac{1}{2} (- \frac{1}{18} + \frac{1}{2} \cdot 1^{- 2})$

Step 14.2.3.17

One to any power is one.

$\frac{- \ln^{2} (3) - 18 (- \frac{\ln^{2} (1)}{2} - \frac{\ln (1)}{2})}{18} - \frac{\ln (3)}{18} + \frac{1}{2} (- \frac{1}{18} + \frac{1}{2} \cdot 1)$

Step 14.2.3.18

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

$\frac{- \ln^{2} (3) - 18 (- \frac{\ln^{2} (1)}{2} - \frac{\ln (1)}{2})}{18} - \frac{\ln (3)}{18} + \frac{1}{2} (- \frac{1}{18} + \frac{1}{2})$

Step 14.2.3.19

To write $\frac{1}{2}$ as a fraction with a common denominator, multiply by $\frac{9}{9}$ .

$\frac{- \ln^{2} (3) - 18 (- \frac{\ln^{2} (1)}{2} - \frac{\ln (1)}{2})}{18} - \frac{\ln (3)}{18} + \frac{1}{2} (- \frac{1}{18} + \frac{1}{2} \cdot \frac{9}{9})$

Step 14.2.3.20

Write each expression with a common denominator of $18$ , by multiplying each by an appropriate factor of $1$ .

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Step 14.2.3.20.1

Multiply $\frac{1}{2}$ by $\frac{9}{9}$ .

$\frac{- \ln^{2} (3) - 18 (- \frac{\ln^{2} (1)}{2} - \frac{\ln (1)}{2})}{18} - \frac{\ln (3)}{18} + \frac{1}{2} (- \frac{1}{18} + \frac{9}{2 \cdot 9})$

Step 14.2.3.20.2

Multiply $2$ by $9$ .

$\frac{- \ln^{2} (3) - 18 (- \frac{\ln^{2} (1)}{2} - \frac{\ln (1)}{2})}{18} - \frac{\ln (3)}{18} + \frac{1}{2} (- \frac{1}{18} + \frac{9}{18})$

Step 14.2.3.21

Combine the numerators over the common denominator.

$\frac{- \ln^{2} (3) - 18 (- \frac{\ln^{2} (1)}{2} - \frac{\ln (1)}{2})}{18} - \frac{\ln (3)}{18} + \frac{1}{2} \cdot \frac{- 1 + 9}{18}$

Step 14.2.3.22

Add $- 1$ and $9$ .

$\frac{- \ln^{2} (3) - 18 (- \frac{\ln^{2} (1)}{2} - \frac{\ln (1)}{2})}{18} - \frac{\ln (3)}{18} + \frac{1}{2} \cdot \frac{8}{18}$

Step 14.2.3.23

Cancel the common factor of $8$ and $18$ .

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Step 14.2.3.23.1

Factor $2$ out of $8$ .

$\frac{- \ln^{2} (3) - 18 (- \frac{\ln^{2} (1)}{2} - \frac{\ln (1)}{2})}{18} - \frac{\ln (3)}{18} + \frac{1}{2} \cdot \frac{2 (4)}{18}$

Step 14.2.3.23.2

Cancel the common factors.

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Step 14.2.3.23.2.1

Factor $2$ out of $18$ .

$\frac{- \ln^{2} (3) - 18 (- \frac{\ln^{2} (1)}{2} - \frac{\ln (1)}{2})}{18} - \frac{\ln (3)}{18} + \frac{1}{2} \cdot \frac{2 \cdot 4}{2 \cdot 9}$

Step 14.2.3.23.2.2

Cancel the common factor.

$\frac{- \ln^{2} (3) - 18 (- \frac{\ln^{2} (1)}{2} - \frac{\ln (1)}{2})}{18} - \frac{\ln (3)}{18} + \frac{1}{2} \cdot \frac{2 \cdot 4}{2 \cdot 9}$

Step 14.2.3.23.2.3

Rewrite the expression.

$\frac{- \ln^{2} (3) - 18 (- \frac{\ln^{2} (1)}{2} - \frac{\ln (1)}{2})}{18} - \frac{\ln (3)}{18} + \frac{1}{2} \cdot \frac{4}{9}$

Step 14.2.3.24

Multiply $\frac{1}{2}$ by $\frac{4}{9}$ .

$\frac{- \ln^{2} (3) - 18 (- \frac{\ln^{2} (1)}{2} - \frac{\ln (1)}{2})}{18} - \frac{\ln (3)}{18} + \frac{4}{2 \cdot 9}$

Step 14.2.3.25

Multiply $2$ by $9$ .

$\frac{- \ln^{2} (3) - 18 (- \frac{\ln^{2} (1)}{2} - \frac{\ln (1)}{2})}{18} - \frac{\ln (3)}{18} + \frac{4}{18}$

Step 14.2.3.26

Cancel the common factor of $4$ and $18$ .

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Step 14.2.3.26.1

Factor $2$ out of $4$ .

$\frac{- \ln^{2} (3) - 18 (- \frac{\ln^{2} (1)}{2} - \frac{\ln (1)}{2})}{18} - \frac{\ln (3)}{18} + \frac{2 (2)}{18}$

Step 14.2.3.26.2

Cancel the common factors.

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Step 14.2.3.26.2.1

Factor $2$ out of $18$ .

$\frac{- \ln^{2} (3) - 18 (- \frac{\ln^{2} (1)}{2} - \frac{\ln (1)}{2})}{18} - \frac{\ln (3)}{18} + \frac{2 \cdot 2}{2 \cdot 9}$

Step 14.2.3.26.2.2

Cancel the common factor.

$\frac{- \ln^{2} (3) - 18 (- \frac{\ln^{2} (1)}{2} - \frac{\ln (1)}{2})}{18} - \frac{\ln (3)}{18} + \frac{2 \cdot 2}{2 \cdot 9}$

Step 14.2.3.26.2.3

Rewrite the expression.

$\frac{- \ln^{2} (3) - 18 (- \frac{\ln^{2} (1)}{2} - \frac{\ln (1)}{2})}{18} - \frac{\ln (3)}{18} + \frac{2}{9}$

Step 15

Simplify.

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Step 15.1

Combine the numerators over the common denominator.

$\frac{- \ln^{2} (3) - 18 (- \frac{\ln^{2} (1)}{2} - \frac{\ln (1)}{2}) - \ln (3)}{18} + \frac{2}{9}$

Step 15.2

Simplify each term.

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Step 15.2.1

Combine the numerators over the common denominator.

$\frac{- \ln^{2} (3) - 18 \frac{- \ln^{2} (1) - \ln (1)}{2} - \ln (3)}{18} + \frac{2}{9}$

Step 15.2.2

Simplify each term.

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Step 15.2.2.1

The natural logarithm of $1$ is $0$ .

$\frac{- \ln^{2} (3) - 18 \frac{- 0^{2} - \ln (1)}{2} - \ln (3)}{18} + \frac{2}{9}$

Step 15.2.2.2

Raising $0$ to any positive power yields $0$ .

$\frac{- \ln^{2} (3) - 18 \frac{- 0 - \ln (1)}{2} - \ln (3)}{18} + \frac{2}{9}$

Step 15.2.2.3

Multiply $- 1$ by $0$ .

$\frac{- \ln^{2} (3) - 18 \frac{0 - \ln (1)}{2} - \ln (3)}{18} + \frac{2}{9}$

Step 15.2.2.4

The natural logarithm of $1$ is $0$ .

$\frac{- \ln^{2} (3) - 18 \frac{0 - 0}{2} - \ln (3)}{18} + \frac{2}{9}$

Step 15.2.2.5

Multiply $- 1$ by $0$ .

$\frac{- \ln^{2} (3) - 18 \frac{0 + 0}{2} - \ln (3)}{18} + \frac{2}{9}$

Step 15.2.3

Add $0$ and $0$ .

$\frac{- \ln^{2} (3) - 18 (\frac{0}{2}) - \ln (3)}{18} + \frac{2}{9}$

Step 15.2.4

Cancel the common factor of $2$ .

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Step 15.2.4.1

Factor $2$ out of $- 18$ .

$\frac{- \ln^{2} (3) + 2 (- 9) \frac{0}{2} - \ln (3)}{18} + \frac{2}{9}$

Step 15.2.4.2

Cancel the common factor.

$\frac{- \ln^{2} (3) + 2 \cdot - 9 \frac{0}{2} - \ln (3)}{18} + \frac{2}{9}$

Step 15.2.4.3

Rewrite the expression.

$\frac{- \ln^{2} (3) - 9 \cdot 0 - \ln (3)}{18} + \frac{2}{9}$

Step 15.2.5

Multiply $- 9$ by $0$ .

$\frac{- \ln^{2} (3) + 0 - \ln (3)}{18} + \frac{2}{9}$

Step 15.3

Add $- \ln^{2} (3)$ and $0$ .

$\frac{- \ln^{2} (3) - \ln (3)}{18} + \frac{2}{9}$

Step 15.4

Simplify each term.

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Step 15.4.1

Factor $- 1$ out of $- \ln^{2} (3) - \ln (3)$ .

$\frac{- (\ln^{2} (3) + \ln (3))}{18} + \frac{2}{9}$

Step 15.4.2

Move the negative in front of the fraction.

$- \frac{\ln^{2} (3) + \ln (3)}{18} + \frac{2}{9}$

Step 15.5

To write $\frac{2}{9}$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$- \frac{\ln^{2} (3) + \ln (3)}{18} + \frac{2}{9} \cdot \frac{2}{2}$

Step 15.6

Write each expression with a common denominator of $18$ , by multiplying each by an appropriate factor of $1$ .

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Step 15.6.1

Multiply $\frac{2}{9}$ by $\frac{2}{2}$ .

$- \frac{\ln^{2} (3) + \ln (3)}{18} + \frac{2 \cdot 2}{9 \cdot 2}$

Step 15.6.2

Multiply $9$ by $2$ .

$- \frac{\ln^{2} (3) + \ln (3)}{18} + \frac{2 \cdot 2}{18}$

Step 15.7

Combine the numerators over the common denominator.

$\frac{- (\ln^{2} (3) + \ln (3)) + 2 \cdot 2}{18}$

Step 15.8

Simplify the numerator.

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Step 15.8.1

Apply the distributive property.

$\frac{- \ln^{2} (3) - \ln (3) + 2 \cdot 2}{18}$

Step 15.8.2

Multiply $2$ by $2$ .

$\frac{- \ln^{2} (3) - \ln (3) + 4}{18}$

Step 16

The result can be shown in multiple forms.

Exact Form:

$\frac{- \ln^{2} (3) - \ln (3) + 4}{18}$

Decimal Form:

$0.09413548 \dots$