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Beliebt Rechnen >

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

Lösung

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

Lösung

ln (x) + 1 - ln ∣ ln (x) + 1 ∣ + C

Schritte zur Lösung

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

Wende U-Substitution an

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

Wende U-Substitution an

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

Multipliziere aus

\frac{v - 1}{v} : 1 - \frac{1}{v}

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

Wende die Summenregel an:

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

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

\int 1 d v = v

\int \frac{1}{v} d v = ln ∣ v ∣

= v - ln ∣ v ∣

Ersetze zurück

= ln (x) + 1 - ln ∣ ln (x) + 1 ∣

Füge eine Konstante zur Lösung hinzu

= ln (x) + 1 - ln ∣ ln (x) + 1 ∣ + C

Graph

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