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Beliebt Rechnen >

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

Lösung

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

Lösung

\frac{2}{3} ln ∣ x - 1 ∣ + \frac{1}{3} ln ∣ x + 2 ∣ - 1 + C

Schritte zur Lösung

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

Multipliziere mit der Konjugation von

x + 1 : \frac{x ^{2} - 1}{x - 1}

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

Wende die partielle Integration an

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

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

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

Vereinfache

(x + 1) (\frac{1}{3} ln ∣ x - 1 ∣ - \frac{1}{3} ln ∣ x + 2 ∣) - \frac{1}{3} (x ln ∣ x - 1 ∣ - x ln ∣ x + 2 ∣ - ln ∣ x - 1 ∣ - 2 ln ∣ x + 2 ∣ + 3) : \frac{2}{3} ln ∣ x - 1 ∣ + \frac{1}{3} ln ∣ x + 2 ∣ - 1

= \frac{2}{3} ln ∣ x - 1 ∣ + \frac{1}{3} ln ∣ x + 2 ∣ - 1

Füge eine Konstante zur Lösung hinzu

= \frac{2}{3} ln ∣ x - 1 ∣ + \frac{1}{3} ln ∣ x + 2 ∣ - 1 + C

Graph

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