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Beliebt Rechnen >

integral of ((x^2-x+8))/(x^3+2x)

Lösung

\int \frac{( x ^{2} - x + 8 )}{x ^{3} + 2 x} d x

Lösung

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

Schritte zur Lösung

\int \frac{x ^{2} - x + 8}{x ^{3} + 2 x} d x

Ermittle den Partialbruch von

\frac{x ^{2} - x + 8}{x ^{3} + 2 x} : \frac{4}{x} + \frac{- 3 x - 1}{x ^{2} + 2}

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

\int \frac{4}{x} d x = 4 ln ∣ x ∣

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

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

Füge eine Konstante zur Lösung hinzu

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

Graph

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