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

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

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

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

- ln ∣ x - x ln (x) ∣ + C

Solution steps

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

Apply u-substitution

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

Take the constant out:

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

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

Use the common integral:

\int \frac{1}{u} d u = ln (∣ u ∣)

= - ln ∣ u ∣

Substitute back

u = x - x ln (x)

= - ln ∣ x - x ln (x) ∣

Add a constant to the solution

= - ln ∣ x - x ln (x) ∣ + C

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