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Beliebt Rechnen >

integral of (x^2+2x-1)/((x+1)(x+2)(x+3))

Lösung

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

Lösung

- ln ∣ x + 1 ∣ + ln ∣ x + 3 ∣ + ln ∣ x + 2 ∣ + C

Schritte zur Lösung

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

Multipliziere aus

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

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

\int \frac{x ^{2}}{( x + 1 ) ( x + 2 ) ( x + 3 )} d x = \frac{1}{2} ln ∣ x + 1 ∣ - 4 ln ∣ x + 2 ∣ + \frac{9}{2} ln ∣ x + 3 ∣

\int \frac{2 x}{( x + 1 ) ( x + 2 ) ( x + 3 )} d x = - ln ∣ x + 1 ∣ + 4 ln ∣ x + 2 ∣ - 3 ln ∣ x + 3 ∣

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

= \frac{1}{2} ln ∣ x + 1 ∣ - 4 ln ∣ x + 2 ∣ + \frac{9}{2} ln ∣ x + 3 ∣ - ln ∣ x + 1 ∣ + 4 ln ∣ x + 2 ∣ - 3 ln ∣ x + 3 ∣ - (\frac{1}{2} ln ∣ x + 1 ∣ - ln ∣ x + 2 ∣ + \frac{1}{2} ln ∣ x + 3 ∣)

Vereinfache

\frac{1}{2} ln ∣ x + 1 ∣ - 4 ln ∣ x + 2 ∣ + \frac{9}{2} ln ∣ x + 3 ∣ - ln ∣ x + 1 ∣ + 4 ln ∣ x + 2 ∣ - 3 ln ∣ x + 3 ∣ - (\frac{1}{2} ln ∣ x + 1 ∣ - ln ∣ x + 2 ∣ + \frac{1}{2} ln ∣ x + 3 ∣) : - ln ∣ x + 1 ∣ + ln ∣ x + 3 ∣ + ln ∣ x + 2 ∣

= - ln ∣ x + 1 ∣ + ln ∣ x + 3 ∣ + ln ∣ x + 2 ∣

Füge eine Konstante zur Lösung hinzu

= - ln ∣ x + 1 ∣ + ln ∣ x + 3 ∣ + ln ∣ x + 2 ∣ + C

Graph

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