What is the integral of (ln(x))/(x(1+ln(x))^2) ?

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

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integral of (ln(x))/(x(1+ln(x))^2)

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

- \frac{ln ( x )}{1 + ln ( x )} + ln ∣ 1 + ln (x) ∣ + C

Solution steps

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

Apply Integration By Parts

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

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

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

Simplify

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

Add a constant to the solution

= - \frac{ln ( x )}{1 + ln ( x )} + ln ∣ 1 + ln (x) ∣ + C

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