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Beliebt Rechnen >

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

Lösung

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

Lösung

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

Schritte zur Lösung

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

Multipliziere aus

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

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

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

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

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

Vereinfache

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

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

Füge eine Konstante zur Lösung hinzu

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

Graph

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