Calculus Examples

$\int_{2}^{4} \frac{3}{x} d x$

Step 1

Since $3$ is constant with respect to $x$ , move $3$ out of the integral.

$3 \int_{2}^{4} \frac{1}{x} d x$

Step 2

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

$3 (\ln (| x |)]_{2}^{4})$

Step 3

Simplify the answer.

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Step 3.1

Evaluate $\ln (| x |)$ at $4$ and at $2$ .

$3 (\ln (| 4 |) - \ln (| 2 |))$

Step 3.2

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

$3 \ln (\frac{| 4 |}{| 2 |})$

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 $0$ and $4$ is $4$ .

$3 \ln (\frac{4}{| 2 |})$

Step 3.3.2

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

$3 \ln (\frac{4}{2})$

Step 3.3.3

Divide $4$ by $2$ .

$3 \ln (2)$

Step 4

The result can be shown in multiple forms.

Exact Form:

$3 \ln (2)$

Decimal Form:

$2.07944154 \dots$

Step 5