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Beliebt Rechnen >

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

Lösung

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

Lösung

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

Schritte zur Lösung

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

Multipliziere aus

\frac{2 x ^{2} - 3 x + 2}{x ^{3} - 3 x ^{2} + 2 x} : \frac{2 x ^{2}}{x ^{3} - 3 x ^{2} + 2 x} - \frac{3 x}{x ^{3} - 3 x ^{2} + 2 x} + \frac{2}{x ^{3} - 3 x ^{2} + 2 x}

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

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

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

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

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

Vereinfache

2 (- ln ∣ x - 1 ∣ + 2 ln ∣ x - 2 ∣) - 3 (- ln ∣ x - 1 ∣ + ln ∣ x - 2 ∣) + ln ∣ x ∣ - 2 ln ∣ x - 1 ∣ + ln ∣ x - 2 ∣ : - ln ∣ x - 1 ∣ + 2 ln ∣ x - 2 ∣ + ln ∣ x ∣

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

Füge eine Konstante zur Lösung hinzu

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

Graph

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