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Beliebt Rechnen >

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

Lösung

integral

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

Lösung

x - \frac{2}{3} ln ∣ x + 1 ∣ - \frac{1}{3} (- ln \frac{4 x ^{2} - 4 x}{3} + \frac{4}{3} + 23 arctan (\frac{2 x - 1}{3})) + C

Schritte zur Lösung

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

Multipliziere aus

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

\int \frac{x ^{3}}{x ^{3} + 1} d x = - \frac{1}{3} ln ∣ x + 1 ∣ - \frac{1}{6} (- ln \frac{4}{3} + \frac{4 x ^{2} - 4 x}{3} + 23 arctan (\frac{2 x - 1}{3})) + x

\int \frac{1}{x ^{3} + 1} d x = \frac{1}{3} ln ∣ x + 1 ∣ + \frac{1}{6} (- ln \frac{4}{3} + \frac{4 x ^{2} - 4 x}{3} + 23 arctan (\frac{2 x - 1}{3}))

= - \frac{1}{3} ln ∣ x + 1 ∣ - \frac{1}{6} (- ln \frac{4}{3} + \frac{4 x ^{2} - 4 x}{3} + 23 arctan (\frac{2 x - 1}{3})) + x - (\frac{1}{3} ln ∣ x + 1 ∣ + \frac{1}{6} (- ln \frac{4}{3} + \frac{4 x ^{2} - 4 x}{3} + 23 arctan (\frac{2 x - 1}{3})))

Vereinfache

- \frac{1}{3} ln ∣ x + 1 ∣ - \frac{1}{6} (- ln \frac{4}{3} + \frac{4 x ^{2} - 4 x}{3} + 23 arctan (\frac{2 x - 1}{3})) + x - (\frac{1}{3} ln ∣ x + 1 ∣ + \frac{1}{6} (- ln \frac{4}{3} + \frac{4 x ^{2} - 4 x}{3} + 23 arctan (\frac{2 x - 1}{3}))) : x - \frac{2}{3} ln ∣ x + 1 ∣ - \frac{1}{3} (- ln \frac{4 x ^{2} - 4 x}{3} + \frac{4}{3} + 23 arctan (\frac{2 x - 1}{3}))

= x - \frac{2}{3} ln ∣ x + 1 ∣ - \frac{1}{3} (- ln \frac{4 x ^{2} - 4 x}{3} + \frac{4}{3} + 23 arctan (\frac{2 x - 1}{3}))

Füge eine Konstante zur Lösung hinzu

= x - \frac{2}{3} ln ∣ x + 1 ∣ - \frac{1}{3} (- ln \frac{4 x ^{2} - 4 x}{3} + \frac{4}{3} + 23 arctan (\frac{2 x - 1}{3})) + C

Graph

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