integral of ln(sqrt(x-1))

Lösung

\int ln (x - 1) d x

\frac{1}{2} (x ln (x - 1) - ln (x - 1) - x + 1) + C

Schritte zur Lösung

\int ln (x - 1) d x

Vereinfache

ln (x - 1) : \frac{1}{2} ln (x - 1)

= \int \frac{1}{2} ln (x - 1) d x

Entferne die Konstante:

\int a \cdot f (x) d x = a \cdot \int f (x) d x

= \frac{1}{2} \cdot \int ln (x - 1) d x

Wende U-Substitution an

= \frac{1}{2} \cdot \int ln (u) d u

Wende die partielle Integration an

= \frac{1}{2} (u ln (u) - \int 1 d u)

\int 1 d u = u

= \frac{1}{2} (u ln (u) - u)

Setze in

u = x - 1

ein

= \frac{1}{2} ((x - 1) ln (x - 1) - (x - 1))

Multipliziere aus

(x - 1) ln (x - 1) - (x - 1) : x ln (x - 1) - ln (x - 1) - x + 1

= \frac{1}{2} (x ln (x - 1) - ln (x - 1) - x + 1)

Füge eine Konstante zur Lösung hinzu

= \frac{1}{2} (x ln (x - 1) - ln (x - 1) - x + 1) + C

n \to \infty lim (\frac{n}{6 ^{n}})

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