What is the integral of (ln(x)-2)/(xln(x)(ln(x)-1)) ?

The integral of (ln(x)-2)/(xln(x)(ln(x)-1)) is -ln|ln(x)-1|+2ln|ln(x)|+C

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integral of (ln(x)-2)/(xln(x)(ln(x)-1))

\int \frac{ln ( x ) - 2}{x ln ( x ) ( ln ( x ) - 1 )} d x

- ln ∣ ln (x) - 1 ∣ + 2 ln ∣ ln (x) ∣ + C

Solution steps

\int \frac{ln ( x ) - 2}{x ln ( x ) ( ln ( x ) - 1 )} d x

Expand

\frac{ln ( x ) - 2}{x ln ( x ) ( ln ( x ) - 1 )} : \frac{ln ( x )}{x ln ( x ) ( ln ( x ) - 1 )} - \frac{2}{x ln ( x ) ( ln ( x ) - 1 )}

Apply the Sum Rule:

\int f (x) \pm g (x) d x = \int f (x) d x \pm \int g (x) d x

= \int \frac{ln ( x )}{x ln ( x ) ( ln ( x ) - 1 )} d x - \int \frac{2}{x ln ( x ) ( ln ( x ) - 1 )} d x

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

\int \frac{2}{x ln ( x ) ( ln ( x ) - 1 )} d x = 2 (- ln ∣ ln (x) ∣ + ln ∣ ln (x) - 1 ∣)

= ln ∣ ln (x) - 1 ∣ - 2 (- ln ∣ ln (x) ∣ + ln ∣ ln (x) - 1 ∣)

Simplify

ln ∣ ln (x) - 1 ∣ - 2 (- ln ∣ ln (x) ∣ + ln ∣ ln (x) - 1 ∣) : - ln ∣ ln (x) - 1 ∣ + 2 ln ∣ ln (x) ∣

= - ln ∣ ln (x) - 1 ∣ + 2 ln ∣ ln (x) ∣

Add a constant to the solution

= - ln ∣ ln (x) - 1 ∣ + 2 ln ∣ ln (x) ∣ + C

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What is the integral of (ln(x)-2)/(xln(x)(ln(x)-1)) ?
The integral of (ln(x)-2)/(xln(x)(ln(x)-1)) is -ln|ln(x)-1|+2ln|ln(x)|+C