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Beliebt Rechnen >

integral of (x^3+1)^{-1}

Lösung

\int (x^{3} + 1)^{- 1} d x

Lösung

\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})) + C

Schritte zur Lösung

\int (x^{3} + 1)^{- 1} d x

Wende Exponentenregel an:

a^{- 1} = \frac{1}{a}

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

Ermittle den Partialbruch von

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

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

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

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

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

Füge eine Konstante zur Lösung hinzu

= \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})) + C

Graph

Beliebte Beispiele

integral of (x(1-x^2))/(1+x^4)

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

integral of (9+tan^2(x))

\int (9 + tan^{2} (x)) d x

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\int \frac{x}{( x ^{3} + 1 )} d x

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\int arcsin (3 t) d t

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\int \frac{- 2 x}{e ^{x}} d x