Calculus Examples

$\int_{- 10}^{- 2} (\frac{1}{x}) d x$

Step 1

Remove parentheses.

$\int_{- 10}^{- 2} \frac{1}{x} d x$

Step 2

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

$\ln (| x |)]_{- 10}^{- 2}$

Step 3

Simplify the answer.

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Step 3.1

Evaluate $\ln (| x |)$ at $- 2$ and at $- 10$ .

$\ln (| - 2 |) - \ln (| - 10 |)$

Step 3.2

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

$\ln (\frac{| - 2 |}{| - 10 |})$

Step 3.3

Simplify.

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Step 3.3.1

The absolute value is the distance between a number and zero. The distance between $- 2$ and $0$ is $2$ .

$\ln (\frac{2}{| - 10 |})$

Step 3.3.2

The absolute value is the distance between a number and zero. The distance between $- 10$ and $0$ is $10$ .

$\ln (\frac{2}{10})$

Step 3.3.3

Cancel the common factor of $2$ and $10$ .

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Step 3.3.3.1

Factor $2$ out of $2$ .

$\ln (\frac{2 (1)}{10})$

Step 3.3.3.2

Cancel the common factors.

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Step 3.3.3.2.1

Factor $2$ out of $10$ .

$\ln (\frac{2 \cdot 1}{2 \cdot 5})$

Step 3.3.3.2.2

Cancel the common factor.

$\ln (\frac{2 \cdot 1}{2 \cdot 5})$

Step 3.3.3.2.3

Rewrite the expression.

$\ln (\frac{1}{5})$

Step 4

The result can be shown in multiple forms.

Exact Form:

$\ln (\frac{1}{5})$

Decimal Form:

$- 1.60943791 \dots$

Step 5