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

Lösung

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

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

Schritte zur Lösung

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

Wende die partielle Integration an

= \frac{1}{( 1 - x ) ^{2}} - \int - \frac{3 x - 1}{x ( x - 1 ) ^{3}} d x

\int - \frac{3 x - 1}{x ( x - 1 ) ^{3}} d x = \frac{1}{( x - 1 ) ^{2}} - ln ∣ x ∣ + ln ∣ x - 1 ∣ + \frac{1}{x - 1}

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

Vereinfache

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

Füge eine Konstante zur Lösung hinzu

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

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