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

Lösung

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

ln ∣ x ∣ + ln ∣ ln (x) ∣ + C

Schritte zur Lösung

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

Multipliziere aus

\frac{ln ( x ) + 1}{x ln ( x )} : \frac{1}{x} + \frac{1}{x ln ( x )}

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

Wende die Summenregel an:

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

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

\int \frac{1}{x} d x = ln ∣ x ∣

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

= ln ∣ x ∣ + ln ∣ ln (x) ∣

Füge eine Konstante zur Lösung hinzu

= ln ∣ x ∣ + ln ∣ ln (x) ∣ + C

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