Calculus Examples

$\int_{2}^{3} \frac{3}{5 x} d x$

Step 1

Since $\frac{3}{5}$ is constant with respect to $x$ , move $\frac{3}{5}$ out of the integral.

$\frac{3}{5} \int_{2}^{3} \frac{1}{x} d x$

Step 2

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

$\frac{3}{5} \ln (| x |)]_{2}^{3}$

Step 3

Simplify the answer.

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Step 3.1

Evaluate $\ln (| x |)$ at $3$ and at $2$ .

$\frac{3}{5} (\ln (| 3 |) - \ln (| 2 |))$

Step 3.2

Simplify.

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Step 3.2.1

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

$\frac{3}{5} \ln (\frac{| 3 |}{| 2 |})$

Step 3.2.2

Combine $\frac{3}{5}$ and $\ln (\frac{| 3 |}{| 2 |})$ .

$\frac{3 \ln (\frac{| 3 |}{| 2 |})}{5}$

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 $3$ is $3$ .

$\frac{3 \ln (\frac{3}{| 2 |})}{5}$

Step 3.3.2

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

$\frac{3 \ln (\frac{3}{2})}{5}$

Step 4

The result can be shown in multiple forms.

Exact Form:

$\frac{3 \ln (\frac{3}{2})}{5}$

Decimal Form:

$0.24327906 \dots$

Step 5