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Beliebt Rechnen >

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

Lösung

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

Ermittle den Partialbruch von

\frac{1}{( x + 1 ) ( x ^{2} - x + 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^2+2x-5)/(x^3-x^2-4x+4)

\int \frac{x ^{2} + 2 x - 5}{x ^{3} - x ^{2} - 4 x + 4} d x

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\int t (t^{2} + t - 1) d t

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

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\int n e^{- n^{2}}

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\int \frac{1}{1 + tan ( u )} d u