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Beliebt Rechnen >

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

Lösung

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

Lösung

\frac{2}{9} ln ∣ x + 1 ∣ - \frac{1}{3 ( x + 1 )} + \frac{1}{9} (- ln x^{2} + 2 - \frac{1}{2} arctan (\frac{x}{2})) + C

Schritte zur Lösung

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

Ermittle den Partialbruch von

\frac{1}{( x + 1 ) ^{2} ( x ^{2} + 2 )} : \frac{2}{9 ( x + 1 )} + \frac{1}{3 ( x + 1 ) ^{2}} + \frac{- 2 x - 1}{9 ( x ^{2} + 2 )}

= \int \frac{2}{9 ( x + 1 )} + \frac{1}{3 ( x + 1 ) ^{2}} + \frac{- 2 x - 1}{9 ( x ^{2} + 2 )} 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{2}{9 ( x + 1 )} d x + \int \frac{1}{3 ( x + 1 ) ^{2}} d x + \int \frac{- 2 x - 1}{9 ( x ^{2} + 2 )} d x

\int \frac{2}{9 ( x + 1 )} d x = \frac{2}{9} ln ∣ x + 1 ∣

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

\int \frac{- 2 x - 1}{9 ( x ^{2} + 2 )} d x = \frac{1}{9} (- ln x^{2} + 2 - \frac{1}{2} arctan (\frac{x}{2}))

= \frac{2}{9} ln ∣ x + 1 ∣ - \frac{1}{3 ( x + 1 )} + \frac{1}{9} (- ln x^{2} + 2 - \frac{1}{2} arctan (\frac{x}{2}))

Füge eine Konstante zur Lösung hinzu

= \frac{2}{9} ln ∣ x + 1 ∣ - \frac{1}{3 ( x + 1 )} + \frac{1}{9} (- ln x^{2} + 2 - \frac{1}{2} arctan (\frac{x}{2})) + C

Graph

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