Calculus Examples

$\int \frac{1}{x \ln (x) \ln (\ln (x))} d x$

Step 1

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

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Step 1.1

Let $u_{1} = \ln (x)$ . Find $\frac{d u_{1}}{d x}$ .

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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

Rewrite the problem using $u_{1}$ and $d u_{1}$ .

$\int \frac{1}{u_{1} \ln (u_{1})} d u_{1}$

Step 2

Let $u_{2} = \ln (u_{1})$ . Then $d u_{2} = \frac{1}{u_{1}} d u_{1}$ , so $u_{1} d u_{2} = d u_{1}$ . Rewrite using $u_{2}$ and $d$ $u_{2}$ .

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Step 2.1

Let $u_{2} = \ln (u_{1})$ . Find $\frac{d u_{2}}{d u_{1}}$ .

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Step 2.1.1

Differentiate $\ln (u_{1})$ .

$\frac{d}{d u_{1}} [\ln (u_{1})]$

Step 2.1.2

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

$\frac{1}{u_{1}}$

Step 2.2

Rewrite the problem using $u_{2}$ and $d u_{2}$ .

$\int \frac{1}{u_{2}} d u_{2}$

Step 3

The integral of $\frac{1}{u_{2}}$ with respect to $u_{2}$ is $\ln (| u_{2} |)$ .

$\ln (| u_{2} |) + C$

Step 4

Substitute back in for each integration substitution variable.

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Step 4.1

Replace all occurrences of $u_{2}$ with $\ln (u_{1})$ .

$\ln (| \ln (u_{1}) |) + C$

Step 4.2

Replace all occurrences of $u_{1}$ with $\ln (x)$ .

$\ln (| \ln (\ln (x)) |) + C$