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

Lösung

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

1 + ln (x) - ln ∣ 1 + ln (x) ∣ + C

Schritte zur Lösung

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

Wende U-Substitution an

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

= 1 + ln (x) - ln ∣ 1 + ln (x) ∣

Füge eine Konstante zur Lösung hinzu

= 1 + ln (x) - ln ∣ 1 + ln (x) ∣ + C

\int sec (5 x + 5) tan (5 x + 5) d x

\int sec^{2} (8 x) tan^{2} (8 x) d x

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\int \frac{1}{9 x ^{2} - 6 x + 7} d x