Calculus Examples

$\int_{e}^{e^{2}} \frac{1}{x \cdot \ln (x)} d x$

Step 1

Let $u = \ln (x)$ . Then $d u = \frac{1}{x} d x$ , so $x d u = d x$ . Rewrite using $u$ and $d$ $u$ .

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Let $u = \ln (x)$ . Find $\frac{d u}{d x}$ .

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Differentiate $\ln (x)$ .

$\frac{d}{d x} [\ln (x)]$

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

$\frac{1}{x}$

Substitute the lower limit in for $x$ in $u = \ln (x)$ .

$u_{lower} = \ln (e)$

The natural logarithm of $e$ is $1$ .

$u_{lower} = 1$

Substitute the upper limit in for $x$ in $u = \ln (x)$ .

$u_{upper} = \ln (e^{2})$

Simplify.

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Use logarithm rules to move $2$ out of the exponent.

$u_{upper} = 2 \ln (e)$

The natural logarithm of $e$ is $1$ .

$u_{upper} = 2 \cdot 1$

Multiply $2$ by $1$ .

$u_{upper} = 2$

The values found for $u_{lower}$ and $u_{upper}$ will be used to evaluate the definite integral.

$u_{lower} = 1 u_{upper} = 2$

Rewrite the problem using $u$ , $d u$ , and the new limits of integration.

$\int_{1}^{2} \frac{1}{u} d u$

Step 2

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

$\ln (| u |)]_{1}^{2}$

Step 3

Evaluate $\ln (| u |)$ at $2$ and at $1$ .

$\ln (| 2 |) - \ln (| 1 |)$

Step 4

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

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

Step 5

Simplify.

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

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

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

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

Divide $2$ by $1$ .

$\ln (2)$

Step 6

The result can be shown in multiple forms.

Exact Form:

$\ln (2)$

Decimal Form:

$0.69314718 \dots$