What is the integral of 1/(xln(x)-x) ?

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

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integral of 1/(xln(x)-x)

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

ln ∣ ln (x) - 1 ∣ + C

Solution steps

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

Apply u-substitution

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

Apply u-substitution

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

Use the common integral:

\int \frac{1}{v} d v = ln (∣ v ∣)

= ln ∣ v ∣

Substitute back

= ln ∣ ln (x) - 1 ∣

Add a constant to the solution

= ln ∣ ln (x) - 1 ∣ + C

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