Calculus Examples

$\int_{1}^{3} \frac{1}{x} d x$

Step 1

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

$\ln (| x |)]_{1}^{3}$

Step 2

Simplify the answer.

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Step 2.1

Evaluate $\ln (| x |)$ at $3$ and at $1$ .

$\ln (| 3 |) - \ln (| 1 |)$

Step 2.2

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

$\ln (\frac{| 3 |}{| 1 |})$

Step 2.3

Simplify.

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Step 2.3.1

The absolute value is the distance between a number and zero. The distance between $0$ and $3$ is $3$ .

$\ln (\frac{3}{| 1 |})$

Step 2.3.2

The absolute value is the distance between a number and zero. The distance between $0$ and $1$ is $1$ .

$\ln (\frac{3}{1})$

Step 2.3.3

Divide $3$ by $1$ .

$\ln (3)$

Step 3

The result can be shown in multiple forms.

Exact Form:

$\ln (3)$

Decimal Form:

$1.09861228 \dots$

Step 4