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

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

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

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

- ln ∣ 1 - ln (x) ∣ + C

Solution steps

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

Apply u-substitution

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

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

Apply u-substitution

: \int - \frac{1}{v} d v

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

Take the constant out:

\int a \cdot f (x) d x = a \cdot \int f (x) d x

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

Use the common integral:

\int \frac{1}{v} d v = ln (∣ v ∣)

= - ln ∣ v ∣

Substitute back

= - ln ∣ 1 - ln (x) ∣

Add a constant to the solution

= - ln ∣ 1 - ln (x) ∣ + C

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