Calculus Examples

$\int_{1}^{9} \frac{1}{2 x} d x$

Step 1

Since $\frac{1}{2}$ is constant with respect to $x$ , move $\frac{1}{2}$ out of the integral.

$\frac{1}{2} \int_{1}^{9} \frac{1}{x} d x$

Step 2

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

$\frac{1}{2} \ln (| x |)]_{1}^{9}$

Step 3

Simplify the answer.

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Step 3.1

Evaluate $\ln (| x |)$ at $9$ and at $1$ .

$\frac{1}{2} (\ln (| 9 |) - \ln (| 1 |))$

Step 3.2

Simplify.

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Step 3.2.1

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

$\frac{1}{2} \ln (\frac{| 9 |}{| 1 |})$

Step 3.2.2

Combine $\frac{1}{2}$ and $\ln (\frac{| 9 |}{| 1 |})$ .

$\frac{\ln (\frac{| 9 |}{| 1 |})}{2}$

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 $9$ is $9$ .

$\frac{\ln (\frac{9}{| 1 |})}{2}$

Step 3.3.2

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

$\frac{\ln (\frac{9}{1})}{2}$

Step 3.3.3

Divide $9$ by $1$ .

$\frac{\ln (9)}{2}$

Step 3.3.4

Rewrite $\ln (9)$ as $\ln (3^{2})$ .

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

Step 3.3.5

Expand $\ln (3^{2})$ by moving $2$ outside the logarithm.

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

Step 3.3.6

Cancel the common factor of $2$ .

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Step 3.3.6.1

Cancel the common factor.

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

Step 3.3.6.2

Divide $\ln (3)$ by $1$ .

$\ln (3)$

Step 4

The result can be shown in multiple forms.

Exact Form:

$\ln (3)$

Decimal Form:

$1.09861228 \dots$

Step 5