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Beliebt Rechnen >

integral of (9x^3-3x+1)/(x^3-x^2)

Lösung

\int \frac{9 x ^{3} - 3 x + 1}{x ^{3} - x ^{2}} d x

Lösung

\frac{9 x ^{2}}{x - 1} + 7 ln ∣ x - 1 ∣ - \frac{9}{x - 1} + 2 ln ∣ x ∣ + \frac{1}{x} + C

Schritte zur Lösung

\int \frac{9 x ^{3} - 3 x + 1}{x ^{3} - x ^{2}} d x

Multipliziere aus

\frac{9 x ^{3} - 3 x + 1}{x ^{3} - x ^{2}} : \frac{9 x ^{3}}{x ^{3} - x ^{2}} - \frac{3 x}{x ^{3} - x ^{2}} + \frac{1}{x ^{3} - x ^{2}}

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

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

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

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

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

Vereinfache

9 (\frac{x ^{2}}{x - 1} + ln ∣ x - 1 ∣ - \frac{1}{x - 1}) - 3 (- ln ∣ x ∣ + ln ∣ x - 1 ∣) - ln ∣ x ∣ + \frac{1}{x} + ln ∣ x - 1 ∣ : \frac{9 x ^{2}}{x - 1} + 7 ln ∣ x - 1 ∣ - \frac{9}{x - 1} + 2 ln ∣ x ∣ + \frac{1}{x}

= \frac{9 x ^{2}}{x - 1} + 7 ln ∣ x - 1 ∣ - \frac{9}{x - 1} + 2 ln ∣ x ∣ + \frac{1}{x}

Füge eine Konstante zur Lösung hinzu

= \frac{9 x ^{2}}{x - 1} + 7 ln ∣ x - 1 ∣ - \frac{9}{x - 1} + 2 ln ∣ x ∣ + \frac{1}{x} + C

Graph

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