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Beliebt Rechnen >

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

Lösung

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

Lösung

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

Schritte zur Lösung

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

Multipliziere aus

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

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

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

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

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

Vereinfache

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

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

Füge eine Konstante zur Lösung hinzu

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

Graph

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