Calculus Examples

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

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

Rewrite the problem using $u$ and $d u$ .

$\int \ln (u) d u$

Step 2

Integrate by parts using the formula $\int w d v = w v - \int v d w$ , where $w = \ln (u)$ and $dv = 1$ .

$\ln (u) u - \int u \frac{1}{u} d u$

Step 3

Simplify.

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Step 3.1

Combine $u$ and $\frac{1}{u}$ .

$\ln (u) u - \int \frac{u}{u} d u$

Step 3.2

Cancel the common factor of $u$ .

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Step 3.2.1

Cancel the common factor.

$\ln (u) u - \int \frac{u}{u} d u$

Step 3.2.2

Rewrite the expression.

$\ln (u) u - \int d u$

Step 4

Apply the constant rule.

$\ln (u) u - (u + C)$

Step 5

Simplify.

$\ln (u) u - u + C$

Step 6

Replace all occurrences of $u$ with $\ln (x)$ .

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