Calculus Examples

$\int_{1}^{2} (9 x^{2} - 4 x + 1) \ln (x) d x$

Step 1

Simplify.

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Step 1.1

Apply the distributive property.

$\int_{1}^{2} 9 x^{2} \ln (x) - 4 x \ln (x) + 1 \ln (x) d x$

Step 1.2

Simplify.

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Step 1.2.1

Simplify $9 \ln (x)$ by moving $9$ inside the logarithm.

$\int_{1}^{2} x^{2} \ln (x^{9}) - 4 x \ln (x) + 1 \ln (x) d x$

Step 1.2.2

Simplify $- 4 \ln (x)$ by moving $4$ inside the logarithm.

$\int_{1}^{2} x^{2} \ln (x^{9}) + x (- \ln (x^{4})) + 1 \ln (x) d x$

Step 1.2.3

Multiply $\ln (x)$ by $1$ .

$\int_{1}^{2} x^{2} \ln (x^{9}) + x (- \ln (x^{4})) + \ln (x) d x$

Step 1.3

Rewrite using the commutative property of multiplication.

$\int_{1}^{2} x^{2} \ln (x^{9}) - x \ln (x^{4}) + \ln (x) d x$

Step 2

Rewrite $x^{2} \ln (x^{9}) - x \ln (x^{4}) + \ln (x)$ as $x^{2} (9 \ln (x)) - x (4 \ln (x)) + \ln (x)$ .

$\int_{1}^{2} x^{2} (9 \ln (x)) - x (4 \ln (x)) + \ln (x) d x$

Step 3

Split the single integral into multiple integrals.

$\int_{1}^{2} x^{2} (9 \ln (x)) d x + \int_{1}^{2} - x (4 \ln (x)) d x + \int_{1}^{2} \ln (x) d x$

Step 4

Multiply $4$ by $- 1$ .

$\int_{1}^{2} x^{2} (9 \ln (x)) d x + \int_{1}^{2} - 4 x \ln (x) d x + \int_{1}^{2} \ln (x) d x$

Step 5

Since $9$ is constant with respect to $x$ , move $9$ out of the integral.

$9 \int_{1}^{2} x^{2} (\ln (x)) d x + \int_{1}^{2} - 4 x \ln (x) d x + \int_{1}^{2} \ln (x) d x$

Step 6

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

$9 (\ln (x) (\frac{1}{3} x^{3})]_{1}^{2} - \int_{1}^{2} \frac{1}{3} x^{3} \frac{1}{x} d x) + \int_{1}^{2} - 4 x \ln (x) d x + \int_{1}^{2} \ln (x) d x$

Step 7

Simplify.

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Step 7.1

Combine $\frac{1}{3}$ and $x^{3}$ .

$9 (\ln (x) \frac{x^{3}}{3}]_{1}^{2} - \int_{1}^{2} \frac{1}{3} x^{3} \frac{1}{x} d x) + \int_{1}^{2} - 4 x \ln (x) d x + \int_{1}^{2} \ln (x) d x$

Step 7.2

Combine $\ln (x)$ and $\frac{x^{3}}{3}$ .

$9 (\frac{\ln (x) x^{3}}{3}]_{1}^{2} - \int_{1}^{2} \frac{1}{3} x^{3} \frac{1}{x} d x) + \int_{1}^{2} - 4 x \ln (x) d x + \int_{1}^{2} \ln (x) d x$

Step 7.3

Combine $\frac{1}{3}$ and $x^{3}$ .

$9 (\frac{\ln (x) x^{3}}{3}]_{1}^{2} - \int_{1}^{2} \frac{x^{3}}{3} \cdot \frac{1}{x} d x) + \int_{1}^{2} - 4 x \ln (x) d x + \int_{1}^{2} \ln (x) d x$

Step 7.4

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

$9 (\frac{\ln (x) x^{3}}{3}]_{1}^{2} - \int_{1}^{2} \frac{x^{3}}{3 x} d x) + \int_{1}^{2} - 4 x \ln (x) d x + \int_{1}^{2} \ln (x) d x$

Step 7.5

Cancel the common factor of $x^{3}$ and $x$ .

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Step 7.5.1

Factor $x$ out of $x^{3}$ .

$9 (\frac{\ln (x) x^{3}}{3}]_{1}^{2} - \int_{1}^{2} \frac{x \cdot x^{2}}{3 x} d x) + \int_{1}^{2} - 4 x \ln (x) d x + \int_{1}^{2} \ln (x) d x$

Step 7.5.2

Cancel the common factors.

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Step 7.5.2.1

Factor $x$ out of $3 x$ .

$9 (\frac{\ln (x) x^{3}}{3}]_{1}^{2} - \int_{1}^{2} \frac{x \cdot x^{2}}{x \cdot 3} d x) + \int_{1}^{2} - 4 x \ln (x) d x + \int_{1}^{2} \ln (x) d x$

Step 7.5.2.2

Cancel the common factor.

$9 (\frac{\ln (x) x^{3}}{3}]_{1}^{2} - \int_{1}^{2} \frac{x \cdot x^{2}}{x \cdot 3} d x) + \int_{1}^{2} - 4 x \ln (x) d x + \int_{1}^{2} \ln (x) d x$

Step 7.5.2.3

Rewrite the expression.

$9 (\frac{\ln (x) x^{3}}{3}]_{1}^{2} - \int_{1}^{2} \frac{x^{2}}{3} d x) + \int_{1}^{2} - 4 x \ln (x) d x + \int_{1}^{2} \ln (x) d x$

Step 8

Since $\frac{1}{3}$ is constant with respect to $x$ , move $\frac{1}{3}$ out of the integral.

$9 (\frac{\ln (x) x^{3}}{3}]_{1}^{2} - (\frac{1}{3} \int_{1}^{2} x^{2} d x)) + \int_{1}^{2} - 4 x \ln (x) d x + \int_{1}^{2} \ln (x) d x$

Step 9

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

$9 (\frac{\ln (x) x^{3}}{3}]_{1}^{2} - \frac{1}{3} \frac{1}{3} x^{3}]_{1}^{2}) + \int_{1}^{2} - 4 x \ln (x) d x + \int_{1}^{2} \ln (x) d x$

Step 10

Simplify.

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Step 10.1

Combine $\frac{1}{3}$ and $x^{3}$ .

$9 (\frac{\ln (x) x^{3}}{3}]_{1}^{2} - \frac{1}{3} \frac{x^{3}}{3}]_{1}^{2}) + \int_{1}^{2} - 4 x \ln (x) d x + \int_{1}^{2} \ln (x) d x$

Step 10.2

Combine $\frac{x^{3}}{3}]_{1}^{2}$ and $\frac{1}{3}$ .

$9 (\frac{\ln (x) x^{3}}{3}]_{1}^{2} - \frac{\frac{x^{3}}{3}]_{1}^{2}}{3}) + \int_{1}^{2} - 4 x \ln (x) d x + \int_{1}^{2} \ln (x) d x$

Step 11

Since $- 4$ is constant with respect to $x$ , move $- 4$ out of the integral.

$9 (\frac{\ln (x) x^{3}}{3}]_{1}^{2} - \frac{\frac{x^{3}}{3}]_{1}^{2}}{3}) - 4 \int_{1}^{2} x \ln (x) d x + \int_{1}^{2} \ln (x) d x$

Step 12

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

$9 (\frac{\ln (x) x^{3}}{3}]_{1}^{2} - \frac{\frac{x^{3}}{3}]_{1}^{2}}{3}) - 4 (\ln (x) (\frac{1}{2} x^{2})]_{1}^{2} - \int_{1}^{2} \frac{1}{2} x^{2} \frac{1}{x} d x) + \int_{1}^{2} \ln (x) d x$

Step 13

Simplify.

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Step 13.1

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

$9 (\frac{\ln (x) x^{3}}{3}]_{1}^{2} - \frac{\frac{x^{3}}{3}]_{1}^{2}}{3}) - 4 (\ln (x) \frac{x^{2}}{2}]_{1}^{2} - \int_{1}^{2} \frac{1}{2} x^{2} \frac{1}{x} d x) + \int_{1}^{2} \ln (x) d x$

Step 13.2

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

$9 (\frac{\ln (x) x^{3}}{3}]_{1}^{2} - \frac{\frac{x^{3}}{3}]_{1}^{2}}{3}) - 4 (\frac{\ln (x) x^{2}}{2}]_{1}^{2} - \int_{1}^{2} \frac{1}{2} x^{2} \frac{1}{x} d x) + \int_{1}^{2} \ln (x) d x$

Step 13.3

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

$9 (\frac{\ln (x) x^{3}}{3}]_{1}^{2} - \frac{\frac{x^{3}}{3}]_{1}^{2}}{3}) - 4 (\frac{\ln (x) x^{2}}{2}]_{1}^{2} - \int_{1}^{2} \frac{x^{2}}{2} \cdot \frac{1}{x} d x) + \int_{1}^{2} \ln (x) d x$

Step 13.4

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

$9 (\frac{\ln (x) x^{3}}{3}]_{1}^{2} - \frac{\frac{x^{3}}{3}]_{1}^{2}}{3}) - 4 (\frac{\ln (x) x^{2}}{2}]_{1}^{2} - \int_{1}^{2} \frac{x^{2}}{2 x} d x) + \int_{1}^{2} \ln (x) d x$

Step 13.5

Cancel the common factor of $x^{2}$ and $x$ .

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Step 13.5.1

Factor $x$ out of $x^{2}$ .

$9 (\frac{\ln (x) x^{3}}{3}]_{1}^{2} - \frac{\frac{x^{3}}{3}]_{1}^{2}}{3}) - 4 (\frac{\ln (x) x^{2}}{2}]_{1}^{2} - \int_{1}^{2} \frac{x \cdot x}{2 x} d x) + \int_{1}^{2} \ln (x) d x$

Step 13.5.2

Cancel the common factors.

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Step 13.5.2.1

Factor $x$ out of $2 x$ .

$9 (\frac{\ln (x) x^{3}}{3}]_{1}^{2} - \frac{\frac{x^{3}}{3}]_{1}^{2}}{3}) - 4 (\frac{\ln (x) x^{2}}{2}]_{1}^{2} - \int_{1}^{2} \frac{x \cdot x}{x \cdot 2} d x) + \int_{1}^{2} \ln (x) d x$

Step 13.5.2.2

Cancel the common factor.

$9 (\frac{\ln (x) x^{3}}{3}]_{1}^{2} - \frac{\frac{x^{3}}{3}]_{1}^{2}}{3}) - 4 (\frac{\ln (x) x^{2}}{2}]_{1}^{2} - \int_{1}^{2} \frac{x \cdot x}{x \cdot 2} d x) + \int_{1}^{2} \ln (x) d x$

Step 13.5.2.3

Rewrite the expression.

$9 (\frac{\ln (x) x^{3}}{3}]_{1}^{2} - \frac{\frac{x^{3}}{3}]_{1}^{2}}{3}) - 4 (\frac{\ln (x) x^{2}}{2}]_{1}^{2} - \int_{1}^{2} \frac{x}{2} d x) + \int_{1}^{2} \ln (x) d x$

Step 14

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

$9 (\frac{\ln (x) x^{3}}{3}]_{1}^{2} - \frac{\frac{x^{3}}{3}]_{1}^{2}}{3}) - 4 (\frac{\ln (x) x^{2}}{2}]_{1}^{2} - (\frac{1}{2} \int_{1}^{2} x d x)) + \int_{1}^{2} \ln (x) d x$

Step 15

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

$9 (\frac{\ln (x) x^{3}}{3}]_{1}^{2} - \frac{\frac{x^{3}}{3}]_{1}^{2}}{3}) - 4 (\frac{\ln (x) x^{2}}{2}]_{1}^{2} - \frac{1}{2} \frac{1}{2} x^{2}]_{1}^{2}) + \int_{1}^{2} \ln (x) d x$

Step 16

Simplify.

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Step 16.1

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

$9 (\frac{\ln (x) x^{3}}{3}]_{1}^{2} - \frac{\frac{x^{3}}{3}]_{1}^{2}}{3}) - 4 (\frac{\ln (x) x^{2}}{2}]_{1}^{2} - \frac{1}{2} \frac{x^{2}}{2}]_{1}^{2}) + \int_{1}^{2} \ln (x) d x$

Step 16.2

Combine $\frac{x^{2}}{2}]_{1}^{2}$ and $\frac{1}{2}$ .

$9 (\frac{\ln (x) x^{3}}{3}]_{1}^{2} - \frac{\frac{x^{3}}{3}]_{1}^{2}}{3}) - 4 (\frac{\ln (x) x^{2}}{2}]_{1}^{2} - \frac{\frac{x^{2}}{2}]_{1}^{2}}{2}) + \int_{1}^{2} \ln (x) d x$

Step 17

Integrate by parts using the formula $\int u d v = u v - \int v d u$ , where $u = \ln (x)$ and $d v = 1$ .

$9 (\frac{\ln (x) x^{3}}{3}]_{1}^{2} - \frac{\frac{x^{3}}{3}]_{1}^{2}}{3}) - 4 (\frac{\ln (x) x^{2}}{2}]_{1}^{2} - \frac{\frac{x^{2}}{2}]_{1}^{2}}{2}) + \ln (x) x]_{1}^{2} - \int_{1}^{2} x \frac{1}{x} d x$

Step 18

Simplify.

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Step 18.1

Combine $x$ and $\frac{1}{x}$ .

Step 18.2

Cancel the common factor of $x$ .

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Step 18.2.1

Cancel the common factor.

Step 18.2.2

Rewrite the expression.

$9 (\frac{\ln (x) x^{3}}{3}]_{1}^{2} - \frac{\frac{x^{3}}{3}]_{1}^{2}}{3}) - 4 (\frac{\ln (x) x^{2}}{2}]_{1}^{2} - \frac{\frac{x^{2}}{2}]_{1}^{2}}{2}) + \ln (x) x]_{1}^{2} - \int_{1}^{2} d x$

Step 19

Apply the constant rule.

$9 (\frac{\ln (x) x^{3}}{3}]_{1}^{2} - \frac{\frac{x^{3}}{3}]_{1}^{2}}{3}) - 4 (\frac{\ln (x) x^{2}}{2}]_{1}^{2} - \frac{\frac{x^{2}}{2}]_{1}^{2}}{2}) + \ln (x) x]_{1}^{2} - (x]_{1}^{2})$

Step 20

Substitute and simplify.

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Step 20.1

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

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

Step 20.2

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

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

Step 20.3

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

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

Step 20.4

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

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

Step 20.5

Evaluate $\ln (x) x$ at $2$ and at $1$ .

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

Step 20.6

Evaluate $x$ at $2$ and at $1$ .

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

Step 20.7

Simplify.

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Step 20.7.1

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

$9 (\frac{\ln (2) \cdot 8}{3} - \frac{\ln (1) \cdot 1^{3}}{3} - \frac{(\frac{2^{3}}{3}) - \frac{1^{3}}{3}}{3}) - 4 ((\frac{\ln (2) \cdot 2^{2}}{2}) - \frac{\ln (1) \cdot 1^{2}}{2} - \frac{(\frac{2^{2}}{2}) - \frac{1^{2}}{2}}{2}) + (\ln (2) \cdot 2) - \ln (1) \cdot 1 - ((2) - 1)$

Step 20.7.2

Move $8$ to the left of $\ln (2)$ .

$9 (\frac{8 \cdot \ln (2)}{3} - \frac{\ln (1) \cdot 1^{3}}{3} - \frac{(\frac{2^{3}}{3}) - \frac{1^{3}}{3}}{3}) - 4 ((\frac{\ln (2) \cdot 2^{2}}{2}) - \frac{\ln (1) \cdot 1^{2}}{2} - \frac{(\frac{2^{2}}{2}) - \frac{1^{2}}{2}}{2}) + (\ln (2) \cdot 2) - \ln (1) \cdot 1 - ((2) - 1)$

Step 20.7.3

One to any power is one.

$9 (\frac{8 \ln (2)}{3} - \frac{\ln (1) \cdot 1}{3} - \frac{(\frac{2^{3}}{3}) - \frac{1^{3}}{3}}{3}) - 4 ((\frac{\ln (2) \cdot 2^{2}}{2}) - \frac{\ln (1) \cdot 1^{2}}{2} - \frac{(\frac{2^{2}}{2}) - \frac{1^{2}}{2}}{2}) + (\ln (2) \cdot 2) - \ln (1) \cdot 1 - ((2) - 1)$

Step 20.7.4

Multiply $\ln (1)$ by $1$ .

$9 (\frac{8 \ln (2)}{3} - \frac{\ln (1)}{3} - \frac{(\frac{2^{3}}{3}) - \frac{1^{3}}{3}}{3}) - 4 ((\frac{\ln (2) \cdot 2^{2}}{2}) - \frac{\ln (1) \cdot 1^{2}}{2} - \frac{(\frac{2^{2}}{2}) - \frac{1^{2}}{2}}{2}) + (\ln (2) \cdot 2) - \ln (1) \cdot 1 - ((2) - 1)$

Step 20.7.5

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

$9 (\frac{8 \ln (2)}{3} - \frac{\ln (1)}{3} - \frac{\frac{8}{3} - \frac{1^{3}}{3}}{3}) - 4 ((\frac{\ln (2) \cdot 2^{2}}{2}) - \frac{\ln (1) \cdot 1^{2}}{2} - \frac{(\frac{2^{2}}{2}) - \frac{1^{2}}{2}}{2}) + (\ln (2) \cdot 2) - \ln (1) \cdot 1 - ((2) - 1)$

Step 20.7.6

One to any power is one.

$9 (\frac{8 \ln (2)}{3} - \frac{\ln (1)}{3} - \frac{\frac{8}{3} - \frac{1}{3}}{3}) - 4 ((\frac{\ln (2) \cdot 2^{2}}{2}) - \frac{\ln (1) \cdot 1^{2}}{2} - \frac{(\frac{2^{2}}{2}) - \frac{1^{2}}{2}}{2}) + (\ln (2) \cdot 2) - \ln (1) \cdot 1 - ((2) - 1)$

Step 20.7.7

Combine the numerators over the common denominator.

$9 (\frac{8 \ln (2)}{3} - \frac{\ln (1)}{3} - \frac{\frac{8 - 1}{3}}{3}) - 4 ((\frac{\ln (2) \cdot 2^{2}}{2}) - \frac{\ln (1) \cdot 1^{2}}{2} - \frac{(\frac{2^{2}}{2}) - \frac{1^{2}}{2}}{2}) + (\ln (2) \cdot 2) - \ln (1) \cdot 1 - ((2) - 1)$

Step 20.7.8

Subtract $1$ from $8$ .

$9 (\frac{8 \ln (2)}{3} - \frac{\ln (1)}{3} - \frac{\frac{7}{3}}{3}) - 4 ((\frac{\ln (2) \cdot 2^{2}}{2}) - \frac{\ln (1) \cdot 1^{2}}{2} - \frac{(\frac{2^{2}}{2}) - \frac{1^{2}}{2}}{2}) + (\ln (2) \cdot 2) - \ln (1) \cdot 1 - ((2) - 1)$

Step 20.7.9

Rewrite $\frac{\frac{7}{3}}{3}$ as a product.

$9 (\frac{8 \ln (2)}{3} - \frac{\ln (1)}{3} - (\frac{7}{3} \cdot \frac{1}{3})) - 4 ((\frac{\ln (2) \cdot 2^{2}}{2}) - \frac{\ln (1) \cdot 1^{2}}{2} - \frac{(\frac{2^{2}}{2}) - \frac{1^{2}}{2}}{2}) + (\ln (2) \cdot 2) - \ln (1) \cdot 1 - ((2) - 1)$

Step 20.7.10

Multiply $\frac{7}{3}$ by $\frac{1}{3}$ .

$9 (\frac{8 \ln (2)}{3} - \frac{\ln (1)}{3} - \frac{7}{3 \cdot 3}) - 4 ((\frac{\ln (2) \cdot 2^{2}}{2}) - \frac{\ln (1) \cdot 1^{2}}{2} - \frac{(\frac{2^{2}}{2}) - \frac{1^{2}}{2}}{2}) + (\ln (2) \cdot 2) - \ln (1) \cdot 1 - ((2) - 1)$

Step 20.7.11

Multiply $3$ by $3$ .

$9 (\frac{8 \ln (2)}{3} - \frac{\ln (1)}{3} - \frac{7}{9}) - 4 ((\frac{\ln (2) \cdot 2^{2}}{2}) - \frac{\ln (1) \cdot 1^{2}}{2} - \frac{(\frac{2^{2}}{2}) - \frac{1^{2}}{2}}{2}) + (\ln (2) \cdot 2) - \ln (1) \cdot 1 - ((2) - 1)$

Step 20.7.12

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

$9 (\frac{8 \ln (2)}{3} - \frac{\ln (1)}{3} - \frac{7}{9}) - 4 (\frac{\ln (2) \cdot 4}{2} - \frac{\ln (1) \cdot 1^{2}}{2} - \frac{(\frac{2^{2}}{2}) - \frac{1^{2}}{2}}{2}) + (\ln (2) \cdot 2) - \ln (1) \cdot 1 - ((2) - 1)$

Step 20.7.13

Move $4$ to the left of $\ln (2)$ .

$9 (\frac{8 \ln (2)}{3} - \frac{\ln (1)}{3} - \frac{7}{9}) - 4 (\frac{4 \cdot \ln (2)}{2} - \frac{\ln (1) \cdot 1^{2}}{2} - \frac{(\frac{2^{2}}{2}) - \frac{1^{2}}{2}}{2}) + (\ln (2) \cdot 2) - \ln (1) \cdot 1 - ((2) - 1)$

Step 20.7.14

Cancel the common factor of $4$ and $2$ .

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Step 20.7.14.1

Factor $2$ out of $4 \ln (2)$ .

$9 (\frac{8 \ln (2)}{3} - \frac{\ln (1)}{3} - \frac{7}{9}) - 4 (\frac{2 (2 \ln (2))}{2} - \frac{\ln (1) \cdot 1^{2}}{2} - \frac{(\frac{2^{2}}{2}) - \frac{1^{2}}{2}}{2}) + (\ln (2) \cdot 2) - \ln (1) \cdot 1 - ((2) - 1)$

Step 20.7.14.2

Cancel the common factors.

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Step 20.7.14.2.1

Factor $2$ out of $2$ .

$9 (\frac{8 \ln (2)}{3} - \frac{\ln (1)}{3} - \frac{7}{9}) - 4 (\frac{2 (2 \ln (2))}{2 (1)} - \frac{\ln (1) \cdot 1^{2}}{2} - \frac{(\frac{2^{2}}{2}) - \frac{1^{2}}{2}}{2}) + (\ln (2) \cdot 2) - \ln (1) \cdot 1 - ((2) - 1)$

Step 20.7.14.2.2

Cancel the common factor.

$9 (\frac{8 \ln (2)}{3} - \frac{\ln (1)}{3} - \frac{7}{9}) - 4 (\frac{2 (2 \ln (2))}{2 \cdot 1} - \frac{\ln (1) \cdot 1^{2}}{2} - \frac{(\frac{2^{2}}{2}) - \frac{1^{2}}{2}}{2}) + (\ln (2) \cdot 2) - \ln (1) \cdot 1 - ((2) - 1)$

Step 20.7.14.2.3

Rewrite the expression.

$9 (\frac{8 \ln (2)}{3} - \frac{\ln (1)}{3} - \frac{7}{9}) - 4 (\frac{2 \ln (2)}{1} - \frac{\ln (1) \cdot 1^{2}}{2} - \frac{(\frac{2^{2}}{2}) - \frac{1^{2}}{2}}{2}) + (\ln (2) \cdot 2) - \ln (1) \cdot 1 - ((2) - 1)$

Step 20.7.14.2.4

Divide $2 \ln (2)$ by $1$ .

$9 (\frac{8 \ln (2)}{3} - \frac{\ln (1)}{3} - \frac{7}{9}) - 4 (2 \ln (2) - \frac{\ln (1) \cdot 1^{2}}{2} - \frac{(\frac{2^{2}}{2}) - \frac{1^{2}}{2}}{2}) + (\ln (2) \cdot 2) - \ln (1) \cdot 1 - ((2) - 1)$

Step 20.7.15

One to any power is one.

$9 (\frac{8 \ln (2)}{3} - \frac{\ln (1)}{3} - \frac{7}{9}) - 4 (2 \ln (2) - \frac{\ln (1) \cdot 1}{2} - \frac{(\frac{2^{2}}{2}) - \frac{1^{2}}{2}}{2}) + (\ln (2) \cdot 2) - \ln (1) \cdot 1 - ((2) - 1)$

Step 20.7.16

Multiply $\ln (1)$ by $1$ .

$9 (\frac{8 \ln (2)}{3} - \frac{\ln (1)}{3} - \frac{7}{9}) - 4 (2 \ln (2) - \frac{\ln (1)}{2} - \frac{(\frac{2^{2}}{2}) - \frac{1^{2}}{2}}{2}) + (\ln (2) \cdot 2) - \ln (1) \cdot 1 - ((2) - 1)$

Step 20.7.17

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

$9 (\frac{8 \ln (2)}{3} - \frac{\ln (1)}{3} - \frac{7}{9}) - 4 (2 \ln (2) - \frac{\ln (1)}{2} - \frac{\frac{4}{2} - \frac{1^{2}}{2}}{2}) + (\ln (2) \cdot 2) - \ln (1) \cdot 1 - ((2) - 1)$

Step 20.7.18

Cancel the common factor of $4$ and $2$ .

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Step 20.7.18.1

Factor $2$ out of $4$ .

$9 (\frac{8 \ln (2)}{3} - \frac{\ln (1)}{3} - \frac{7}{9}) - 4 (2 \ln (2) - \frac{\ln (1)}{2} - \frac{\frac{2 \cdot 2}{2} - \frac{1^{2}}{2}}{2}) + (\ln (2) \cdot 2) - \ln (1) \cdot 1 - ((2) - 1)$

Step 20.7.18.2

Cancel the common factors.

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Step 20.7.18.2.1

Factor $2$ out of $2$ .

$9 (\frac{8 \ln (2)}{3} - \frac{\ln (1)}{3} - \frac{7}{9}) - 4 (2 \ln (2) - \frac{\ln (1)}{2} - \frac{\frac{2 \cdot 2}{2 (1)} - \frac{1^{2}}{2}}{2}) + (\ln (2) \cdot 2) - \ln (1) \cdot 1 - ((2) - 1)$

Step 20.7.18.2.2

Cancel the common factor.

$9 (\frac{8 \ln (2)}{3} - \frac{\ln (1)}{3} - \frac{7}{9}) - 4 (2 \ln (2) - \frac{\ln (1)}{2} - \frac{\frac{2 \cdot 2}{2 \cdot 1} - \frac{1^{2}}{2}}{2}) + (\ln (2) \cdot 2) - \ln (1) \cdot 1 - ((2) - 1)$

Step 20.7.18.2.3

Rewrite the expression.

$9 (\frac{8 \ln (2)}{3} - \frac{\ln (1)}{3} - \frac{7}{9}) - 4 (2 \ln (2) - \frac{\ln (1)}{2} - \frac{\frac{2}{1} - \frac{1^{2}}{2}}{2}) + (\ln (2) \cdot 2) - \ln (1) \cdot 1 - ((2) - 1)$

Step 20.7.18.2.4

Divide $2$ by $1$ .

$9 (\frac{8 \ln (2)}{3} - \frac{\ln (1)}{3} - \frac{7}{9}) - 4 (2 \ln (2) - \frac{\ln (1)}{2} - \frac{2 - \frac{1^{2}}{2}}{2}) + (\ln (2) \cdot 2) - \ln (1) \cdot 1 - ((2) - 1)$

Step 20.7.19

One to any power is one.

$9 (\frac{8 \ln (2)}{3} - \frac{\ln (1)}{3} - \frac{7}{9}) - 4 (2 \ln (2) - \frac{\ln (1)}{2} - \frac{2 - \frac{1}{2}}{2}) + (\ln (2) \cdot 2) - \ln (1) \cdot 1 - ((2) - 1)$

Step 20.7.20

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

$9 (\frac{8 \ln (2)}{3} - \frac{\ln (1)}{3} - \frac{7}{9}) - 4 (2 \ln (2) - \frac{\ln (1)}{2} - \frac{2 \cdot \frac{2}{2} - \frac{1}{2}}{2}) + (\ln (2) \cdot 2) - \ln (1) \cdot 1 - ((2) - 1)$

Step 20.7.21

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

$9 (\frac{8 \ln (2)}{3} - \frac{\ln (1)}{3} - \frac{7}{9}) - 4 (2 \ln (2) - \frac{\ln (1)}{2} - \frac{\frac{2 \cdot 2}{2} - \frac{1}{2}}{2}) + (\ln (2) \cdot 2) - \ln (1) \cdot 1 - ((2) - 1)$

Step 20.7.22

Combine the numerators over the common denominator.

$9 (\frac{8 \ln (2)}{3} - \frac{\ln (1)}{3} - \frac{7}{9}) - 4 (2 \ln (2) - \frac{\ln (1)}{2} - \frac{\frac{2 \cdot 2 - 1}{2}}{2}) + (\ln (2) \cdot 2) - \ln (1) \cdot 1 - ((2) - 1)$

Step 20.7.23

Simplify the numerator.

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Step 20.7.23.1

Multiply $2$ by $2$ .

$9 (\frac{8 \ln (2)}{3} - \frac{\ln (1)}{3} - \frac{7}{9}) - 4 (2 \ln (2) - \frac{\ln (1)}{2} - \frac{\frac{4 - 1}{2}}{2}) + (\ln (2) \cdot 2) - \ln (1) \cdot 1 - ((2) - 1)$

Step 20.7.23.2

Subtract $1$ from $4$ .

$9 (\frac{8 \ln (2)}{3} - \frac{\ln (1)}{3} - \frac{7}{9}) - 4 (2 \ln (2) - \frac{\ln (1)}{2} - \frac{\frac{3}{2}}{2}) + (\ln (2) \cdot 2) - \ln (1) \cdot 1 - ((2) - 1)$

Step 20.7.24

Rewrite $\frac{\frac{3}{2}}{2}$ as a product.

$9 (\frac{8 \ln (2)}{3} - \frac{\ln (1)}{3} - \frac{7}{9}) - 4 (2 \ln (2) - \frac{\ln (1)}{2} - (\frac{3}{2} \cdot \frac{1}{2})) + (\ln (2) \cdot 2) - \ln (1) \cdot 1 - ((2) - 1)$

Step 20.7.25

Multiply $\frac{3}{2}$ by $\frac{1}{2}$ .

$9 (\frac{8 \ln (2)}{3} - \frac{\ln (1)}{3} - \frac{7}{9}) - 4 (2 \ln (2) - \frac{\ln (1)}{2} - \frac{3}{2 \cdot 2}) + (\ln (2) \cdot 2) - \ln (1) \cdot 1 - ((2) - 1)$

Step 20.7.26

Multiply $2$ by $2$ .

$9 (\frac{8 \ln (2)}{3} - \frac{\ln (1)}{3} - \frac{7}{9}) - 4 (2 \ln (2) - \frac{\ln (1)}{2} - \frac{3}{4}) + (\ln (2) \cdot 2) - \ln (1) \cdot 1 - ((2) - 1)$

Step 20.7.27

Move $2$ to the left of $\ln (2)$ .

$9 (\frac{8 \ln (2)}{3} - \frac{\ln (1)}{3} - \frac{7}{9}) - 4 (2 \ln (2) - \frac{\ln (1)}{2} - \frac{3}{4}) + 2 \cdot \ln (2) - \ln (1) \cdot 1 - ((2) - 1)$

Step 20.7.28

Multiply $- 1$ by $1$ .

$9 (\frac{8 \ln (2)}{3} - \frac{\ln (1)}{3} - \frac{7}{9}) - 4 (2 \ln (2) - \frac{\ln (1)}{2} - \frac{3}{4}) + 2 \ln (2) - \ln (1) - ((2) - 1)$

Step 20.7.29

Subtract $1$ from $2$ .

$9 (\frac{8 \ln (2)}{3} - \frac{\ln (1)}{3} - \frac{7}{9}) - 4 (2 \ln (2) - \frac{\ln (1)}{2} - \frac{3}{4}) + 2 \ln (2) - \ln (1) - 1 \cdot 1$

Step 20.7.30

Multiply $- 1$ by $1$ .

$9 (\frac{8 \ln (2)}{3} - \frac{\ln (1)}{3} - \frac{7}{9}) - 4 (2 \ln (2) - \frac{\ln (1)}{2} - \frac{3}{4}) + 2 \ln (2) - \ln (1) - 1$

Step 21

Simplify.

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Step 21.1

Simplify each term.

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Step 21.1.1

Combine the numerators over the common denominator.

$9 (\frac{8 \ln (2) - \ln (1)}{3} + \frac{- 7}{9}) - 4 (2 \ln (2) - \frac{\ln (1)}{2} - \frac{3}{4}) + 2 \ln (2) - \ln (1) - 1$

Step 21.1.2

Simplify each term.

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Step 21.1.2.1

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

$9 (\frac{8 \ln (2) - 0}{3} + \frac{- 7}{9}) - 4 (2 \ln (2) - \frac{\ln (1)}{2} - \frac{3}{4}) + 2 \ln (2) - \ln (1) - 1$

Step 21.1.2.2

Multiply $- 1$ by $0$ .

$9 (\frac{8 \ln (2) + 0}{3} + \frac{- 7}{9}) - 4 (2 \ln (2) - \frac{\ln (1)}{2} - \frac{3}{4}) + 2 \ln (2) - \ln (1) - 1$

Step 21.1.3

Add $8 \ln (2)$ and $0$ .

$9 (\frac{8 \ln (2)}{3} + \frac{- 7}{9}) - 4 (2 \ln (2) - \frac{\ln (1)}{2} - \frac{3}{4}) + 2 \ln (2) - \ln (1) - 1$

Step 21.1.4

Move the negative in front of the fraction.

$9 (\frac{8 \ln (2)}{3} - \frac{7}{9}) - 4 (2 \ln (2) - \frac{\ln (1)}{2} - \frac{3}{4}) + 2 \ln (2) - \ln (1) - 1$

Step 21.1.5

Apply the distributive property.

$9 \frac{8 \ln (2)}{3} + 9 (- \frac{7}{9}) - 4 (2 \ln (2) - \frac{\ln (1)}{2} - \frac{3}{4}) + 2 \ln (2) - \ln (1) - 1$

Step 21.1.6

Cancel the common factor of $3$ .

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Step 21.1.6.1

Factor $3$ out of $9$ .

$3 (3) \frac{8 \ln (2)}{3} + 9 (- \frac{7}{9}) - 4 (2 \ln (2) - \frac{\ln (1)}{2} - \frac{3}{4}) + 2 \ln (2) - \ln (1) - 1$

Step 21.1.6.2

Cancel the common factor.

$3 \cdot 3 \frac{8 \ln (2)}{3} + 9 (- \frac{7}{9}) - 4 (2 \ln (2) - \frac{\ln (1)}{2} - \frac{3}{4}) + 2 \ln (2) - \ln (1) - 1$

Step 21.1.6.3

Rewrite the expression.

$3 (8 \ln (2)) + 9 (- \frac{7}{9}) - 4 (2 \ln (2) - \frac{\ln (1)}{2} - \frac{3}{4}) + 2 \ln (2) - \ln (1) - 1$

Step 21.1.7

Multiply $8$ by $3$ .

$24 \ln (2) + 9 (- \frac{7}{9}) - 4 (2 \ln (2) - \frac{\ln (1)}{2} - \frac{3}{4}) + 2 \ln (2) - \ln (1) - 1$

Step 21.1.8

Cancel the common factor of $9$ .

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Step 21.1.8.1

Move the leading negative in $- \frac{7}{9}$ into the numerator.

$24 \ln (2) + 9 (\frac{- 7}{9}) - 4 (2 \ln (2) - \frac{\ln (1)}{2} - \frac{3}{4}) + 2 \ln (2) - \ln (1) - 1$

Step 21.1.8.2

Cancel the common factor.

$24 \ln (2) + 9 (\frac{- 7}{9}) - 4 (2 \ln (2) - \frac{\ln (1)}{2} - \frac{3}{4}) + 2 \ln (2) - \ln (1) - 1$

Step 21.1.8.3

Rewrite the expression.

$24 \ln (2) - 7 - 4 (2 \ln (2) - \frac{\ln (1)}{2} - \frac{3}{4}) + 2 \ln (2) - \ln (1) - 1$

Step 21.1.9

Simplify each term.

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Step 21.1.9.1

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

$24 \ln (2) - 7 - 4 (2 \ln (2) - \frac{0}{2} - \frac{3}{4}) + 2 \ln (2) - \ln (1) - 1$

Step 21.1.9.2

Divide $0$ by $2$ .

$24 \ln (2) - 7 - 4 (2 \ln (2) - 0 - \frac{3}{4}) + 2 \ln (2) - \ln (1) - 1$

Step 21.1.9.3

Multiply $- 1$ by $0$ .

$24 \ln (2) - 7 - 4 (2 \ln (2) + 0 - \frac{3}{4}) + 2 \ln (2) - \ln (1) - 1$

Step 21.1.10

Add $2 \ln (2)$ and $0$ .

$24 \ln (2) - 7 - 4 (2 \ln (2) - \frac{3}{4}) + 2 \ln (2) - \ln (1) - 1$

Step 21.1.11

Apply the distributive property.

$24 \ln (2) - 7 - 4 (2 \ln (2)) - 4 (- \frac{3}{4}) + 2 \ln (2) - \ln (1) - 1$

Step 21.1.12

Multiply $2$ by $- 4$ .

$24 \ln (2) - 7 - 8 \ln (2) - 4 (- \frac{3}{4}) + 2 \ln (2) - \ln (1) - 1$

Step 21.1.13

Cancel the common factor of $4$ .

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Step 21.1.13.1

Move the leading negative in $- \frac{3}{4}$ into the numerator.

$24 \ln (2) - 7 - 8 \ln (2) - 4 (\frac{- 3}{4}) + 2 \ln (2) - \ln (1) - 1$

Step 21.1.13.2

Factor $4$ out of $- 4$ .

$24 \ln (2) - 7 - 8 \ln (2) + 4 (- 1) \frac{- 3}{4} + 2 \ln (2) - \ln (1) - 1$

Step 21.1.13.3

Cancel the common factor.

$24 \ln (2) - 7 - 8 \ln (2) + 4 \cdot - 1 \frac{- 3}{4} + 2 \ln (2) - \ln (1) - 1$

Step 21.1.13.4

Rewrite the expression.

$24 \ln (2) - 7 - 8 \ln (2) - - 3 + 2 \ln (2) - \ln (1) - 1$

Step 21.1.14

Multiply $- 1$ by $- 3$ .

$24 \ln (2) - 7 - 8 \ln (2) + 3 + 2 \ln (2) - \ln (1) - 1$

Step 21.1.15

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

$24 \ln (2) - 7 - 8 \ln (2) + 3 + 2 \ln (2) - 0 - 1$

Step 21.1.16

Multiply $- 1$ by $0$ .

$24 \ln (2) - 7 - 8 \ln (2) + 3 + 2 \ln (2) + 0 - 1$

Step 21.2

Subtract $8 \ln (2)$ from $24 \ln (2)$ .

$16 \ln (2) - 7 + 3 + 2 \ln (2) + 0 - 1$

Step 21.3

Add $16 \ln (2)$ and $2 \ln (2)$ .

$18 \ln (2) - 7 + 3 + 0 - 1$

Step 21.4

Add $18 \ln (2)$ and $0$ .

$18 \ln (2) - 7 + 3 - 1$

Step 21.5

Add $- 7$ and $3$ .

$18 \ln (2) - 4 - 1$

Step 21.6

Subtract $1$ from $- 4$ .

$18 \ln (2) - 5$

Step 22

The result can be shown in multiple forms.

Exact Form:

$18 \ln (2) - 5$

Decimal Form:

$7.47664925 \dots$