What is the integral of 1/x*1/(ln(x)) ?

The integral of 1/x*1/(ln(x)) is ln|ln(x)|+C

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integral of 1/x*1/(ln(x))

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

ln ∣ ln (x) ∣ + C

Solution steps

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

Simplify

\frac{1}{x} \cdot \frac{1}{ln ( x )} : \frac{1}{x ln ( x )}

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

Apply u-substitution

= \int \frac{1}{u} d u

Use the common integral:

\int \frac{1}{u} d u = ln (∣ u ∣)

= ln ∣ u ∣

Substitute back

u = ln (x)

= ln ∣ ln (x) ∣

Add a constant to the solution

= ln ∣ ln (x) ∣ + C

What is the integral of 1/x*1/(ln(x)) ?
The integral of 1/x*1/(ln(x)) is ln|ln(x)|+C