Calculus Examples

$\int_{1}^{4} \frac{1}{x} d x$

Step 1

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

$\ln (| x |)]_{1}^{4}$

Step 2

Simplify the answer.

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Step 2.1

Evaluate $\ln (| x |)$ at $4$ and at $1$ .

$\ln (| 4 |) - \ln (| 1 |)$

Step 2.2

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

$\ln (\frac{| 4 |}{| 1 |})$

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 $4$ is $4$ .

$\ln (\frac{4}{| 1 |})$

Step 2.3.2

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

$\ln (\frac{4}{1})$

Step 2.3.3

Divide $4$ by $1$ .

$\ln (4)$

Step 3

The result can be shown in multiple forms.

Exact Form:

$\ln (4)$

Decimal Form:

$1.38629436 \dots$

Step 4