Calculus Examples

$\int_{8}^{9} x + \frac{2}{x} d x$

Step 1

Split the single integral into multiple integrals.

$\int_{8}^{9} x d x + \int_{8}^{9} \frac{2}{x} d x$

Step 2

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

$\frac{1}{2} x^{2}]_{8}^{9} + \int_{8}^{9} \frac{2}{x} d x$

Step 3

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

$\frac{1}{2} x^{2}]_{8}^{9} + 2 \int_{8}^{9} \frac{1}{x} d x$

Step 4

The integral of $\frac{1}{x}$ with respect to $x$ is $\ln (| x |)$ .

$\frac{1}{2} x^{2}]_{8}^{9} + 2 (\ln (| x |)]_{8}^{9})$

Step 5

Simplify the answer.

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Step 5.1

Substitute and simplify.

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Step 5.1.1

Evaluate $\frac{1}{2} x^{2}$ at $9$ and at $8$ .

$(\frac{1}{2} \cdot 9^{2}) - \frac{1}{2} \cdot 8^{2} + 2 (\ln (| x |)]_{8}^{9})$

Step 5.1.2

Evaluate $\ln (| x |)$ at $9$ and at $8$ .

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

Step 5.1.3

Simplify.

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Step 5.1.3.1

Raise $9$ to the power of $2$ .

$\frac{1}{2} \cdot 81 - \frac{1}{2} \cdot 8^{2} + 2 (\ln (| 9 |) - \ln (| 8 |))$

Step 5.1.3.2

Combine $\frac{1}{2}$ and $81$ .

$\frac{81}{2} - \frac{1}{2} \cdot 8^{2} + 2 (\ln (| 9 |) - \ln (| 8 |))$

Step 5.1.3.3

Raise $8$ to the power of $2$ .

$\frac{81}{2} - \frac{1}{2} \cdot 64 + 2 (\ln (| 9 |) - \ln (| 8 |))$

Step 5.1.3.4

Multiply $64$ by $- 1$ .

$\frac{81}{2} - 64 (\frac{1}{2}) + 2 (\ln (| 9 |) - \ln (| 8 |))$

Step 5.1.3.5

Combine $- 64$ and $\frac{1}{2}$ .

$\frac{81}{2} + \frac{- 64}{2} + 2 (\ln (| 9 |) - \ln (| 8 |))$

Step 5.1.3.6

Cancel the common factor of $- 64$ and $2$ .

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Step 5.1.3.6.1

Factor $2$ out of $- 64$ .

$\frac{81}{2} + \frac{2 \cdot - 32}{2} + 2 (\ln (| 9 |) - \ln (| 8 |))$

Step 5.1.3.6.2

Cancel the common factors.

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Step 5.1.3.6.2.1

Factor $2$ out of $2$ .

$\frac{81}{2} + \frac{2 \cdot - 32}{2 (1)} + 2 (\ln (| 9 |) - \ln (| 8 |))$

Step 5.1.3.6.2.2

Cancel the common factor.

$\frac{81}{2} + \frac{2 \cdot - 32}{2 \cdot 1} + 2 (\ln (| 9 |) - \ln (| 8 |))$

Step 5.1.3.6.2.3

Rewrite the expression.

$\frac{81}{2} + \frac{- 32}{1} + 2 (\ln (| 9 |) - \ln (| 8 |))$

Step 5.1.3.6.2.4

Divide $- 32$ by $1$ .

$\frac{81}{2} - 32 + 2 (\ln (| 9 |) - \ln (| 8 |))$

Step 5.1.3.7

To write $- 32$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$\frac{81}{2} - 32 \cdot \frac{2}{2} + 2 (\ln (| 9 |) - \ln (| 8 |))$

Step 5.1.3.8

Combine $- 32$ and $\frac{2}{2}$ .

$\frac{81}{2} + \frac{- 32 \cdot 2}{2} + 2 (\ln (| 9 |) - \ln (| 8 |))$

Step 5.1.3.9

Combine the numerators over the common denominator.

$\frac{81 - 32 \cdot 2}{2} + 2 (\ln (| 9 |) - \ln (| 8 |))$

Step 5.1.3.10

Simplify the numerator.

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Step 5.1.3.10.1

Multiply $- 32$ by $2$ .

$\frac{81 - 64}{2} + 2 (\ln (| 9 |) - \ln (| 8 |))$

Step 5.1.3.10.2

Subtract $64$ from $81$ .

$\frac{17}{2} + 2 (\ln (| 9 |) - \ln (| 8 |))$

Step 5.1.3.11

To write $2 (\ln (| 9 |) - \ln (| 8 |))$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$\frac{17}{2} + 2 (\ln (| 9 |) - \ln (| 8 |)) \cdot \frac{2}{2}$

Step 5.1.3.12

Combine $2 (\ln (| 9 |) - \ln (| 8 |))$ and $\frac{2}{2}$ .

$\frac{17}{2} + \frac{2 (\ln (| 9 |) - \ln (| 8 |)) \cdot 2}{2}$

Step 5.1.3.13

Combine the numerators over the common denominator.

$\frac{17 + 2 (\ln (| 9 |) - \ln (| 8 |)) \cdot 2}{2}$

Step 5.1.3.14

Multiply $2$ by $2$ .

$\frac{17 + 4 (\ln (| 9 |) - \ln (| 8 |))}{2}$

Step 5.2

Use the quotient property of logarithms, $\log_{b} (x) - \log_{b} (y) = \log_{b} (\frac{x}{y})$ .

$\frac{17 + 4 \ln (\frac{| 9 |}{| 8 |})}{2}$

Step 5.3

Simplify.

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Step 5.3.1

The absolute value is the distance between a number and zero. The distance between $0$ and $9$ is $9$ .

$\frac{17 + 4 \ln (\frac{9}{| 8 |})}{2}$

Step 5.3.2

The absolute value is the distance between a number and zero. The distance between $0$ and $8$ is $8$ .

$\frac{17 + 4 \ln (\frac{9}{8})}{2}$

Step 6

The result can be shown in multiple forms.

Exact Form:

$\frac{17 + 4 \ln (\frac{9}{8})}{2}$

Decimal Form:

$8.73556607 \dots$

Step 7