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

The integral of (ln(u))/((1+ln(u))^2) is -(uln(u))/(1+ln(u))+u+C

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

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

- \frac{u ln ( u )}{1 + ln ( u )} + u + C

Solution steps

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

Apply u-substitution

: \int \frac{e ^{v} v}{( 1 + v ) ^{2}} d v

= \int \frac{e ^{v} v}{( 1 + v ) ^{2}} d v

Apply Integration By Parts

: - \frac{e ^{v} v}{1 + v} - \int - e^{v} d v

= - \frac{e ^{v} v}{1 + v} - \int - e^{v} d v

\int - e^{v} d v = - e^{v}

= - \frac{e ^{v} v}{1 + v} - (- e^{v})

Substitute back

v = ln (u)

= - \frac{e ^{l n (u)} ln ( u )}{1 + ln ( u )} - (- e^{l n (u)})

Simplify

- \frac{e ^{l n (u)} ln ( u )}{1 + ln ( u )} - (- e^{l n (u)}) : - \frac{u ln ( u )}{1 + ln ( u )} + u

= - \frac{u ln ( u )}{1 + ln ( u )} + u

Add a constant to the solution

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

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