Calculus Examples

$\int_{e}^{4} \frac{1}{x \ln (x)} d x$

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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Step 1.1

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

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Step 1.1.1

Differentiate $\ln (x)$ .

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

Step 1.1.2

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

$\frac{1}{x}$

Step 1.2

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

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

Step 1.3

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

$u_{lower} = 1$

Step 1.4

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

$u_{upper} = \ln (4)$

Step 1.5

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

$u_{lower} = 1$

$u_{upper} = \ln (4)$

Step 1.6

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

$\int_{1}^{\ln (4)} \frac{1}{u} d u$

Step 2

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

$\ln (| u |)]_{1}^{\ln (4)}$

Step 3

Evaluate $\ln (| u |)$ at $\ln (4)$ and at $1$ .

$\ln (| \ln (4) |) - \ln (| 1 |)$

Step 4

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

$\ln (\frac{| \ln (4) |}{| 1 |})$

Step 5

Simplify.

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Step 5.1

$\ln (4)$ is approximately $1.38629436$ which is positive so remove the absolute value

$\ln (\frac{\ln (4)}{| 1 |})$

Step 5.2

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

$\ln (\frac{\ln (4)}{1})$

Step 5.3

Divide $\ln (4)$ by $1$ .

$\ln (\ln (4))$

Step 6

The result can be shown in multiple forms.

Exact Form:

$\ln (\ln (4))$

Decimal Form:

$0.32663425 \dots$