Calculus Examples

$\int_{1}^{8} \frac{1}{x} d x$

Step 1

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

$\ln (| x |)]_{1}^{8}$

Step 2

Simplify the answer.

Tap for more steps...

Step 2.1

Evaluate $\ln (| x |)$ at $8$ and at $1$ .

$\ln (| 8 |) - \ln (| 1 |)$

Step 2.2

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

$\ln (\frac{| 8 |}{| 1 |})$

Step 2.3

Simplify.

Tap for more steps...

Step 2.3.1

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

$\ln (\frac{8}{| 1 |})$

Step 2.3.2

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

$\ln (\frac{8}{1})$

Step 2.3.3

Divide $8$ by $1$ .

$\ln (8)$

Step 3

The result can be shown in multiple forms.

Exact Form:

$\ln (8)$

Decimal Form:

$2.07944154 \dots$

Step 4