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

Lösung

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

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

Schritte zur Lösung

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

Wende die partielle Integration an

= - \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) ∣)

Vereinfache

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

Füge eine Konstante zur Lösung hinzu

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

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