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Beliebt Rechnen >

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

Lösung

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

Lösung

\frac{x ^{2}}{2} - x + \frac{1}{3} ln ∣ x - 1 ∣ + \frac{8}{3} ln ∣ x + 2 ∣ - \frac{1}{3} (ln \frac{4}{3} + \frac{2}{3} x - ln - \frac{2}{3} + \frac{2}{3} x) + C

Schritte zur Lösung

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

Multipliziere aus

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

\int \frac{x ^{3}}{x ^{2} + x - 2} d x = \frac{x ^{2}}{2} - x + \frac{1}{3} ln ∣ x - 1 ∣ + \frac{8}{3} ln ∣ x + 2 ∣

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

= \frac{x ^{2}}{2} - x + \frac{1}{3} ln ∣ x - 1 ∣ + \frac{8}{3} ln ∣ x + 2 ∣ - \frac{1}{3} (ln \frac{4}{3} + \frac{2}{3} x - ln - \frac{2}{3} + \frac{2}{3} x)

Füge eine Konstante zur Lösung hinzu

= \frac{x ^{2}}{2} - x + \frac{1}{3} ln ∣ x - 1 ∣ + \frac{8}{3} ln ∣ x + 2 ∣ - \frac{1}{3} (ln \frac{4}{3} + \frac{2}{3} x - ln - \frac{2}{3} + \frac{2}{3} x) + C

Graph

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