integral of ln(1/(x-2))

Lösung

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

- x ln (x - 2) + 2 ln (x - 2) + x - 2 + C

Schritte zur Lösung

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

Wende die log Regel an:

lo g_{a} (\frac{1}{x}) = - lo g_{a} (x)

= \int - ln (x - 2) d x

Entferne die Konstante:

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

= - \int ln (x - 2) d x

Wende U-Substitution an

= - \int ln (u) d u

Wende die partielle Integration an

= - (u ln (u) - \int 1 d u)

\int 1 d u = u

= - (u ln (u) - u)

Setze in

u = x - 2

ein

= - ((x - 2) ln (x - 2) - (x - 2))

Vereinfache

- ((x - 2) ln (x - 2) - (x - 2)) : - x ln (x - 2) + 2 ln (x - 2) + x - 2

= - x ln (x - 2) + 2 ln (x - 2) + x - 2

Füge eine Konstante zur Lösung hinzu

= - x ln (x - 2) + 2 ln (x - 2) + x - 2 + C

\int 1 0^{- y} d y

\frac{d}{d x} (2 x^{3} - 15 x^{2} + 36 x - 10)

\int (t - \frac{1}{t}) d t

\frac{d}{d x} (sin (7 x + ln (5 x)))

x \to 0 lim (\frac{1}{x ^{2} ( x + 7 )})