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Beliebt Rechnen >

integral of ((x^2-x+30))/(x^3+5x)

Lösung

\int \frac{( x ^{2} - x + 30 )}{x ^{3} + 5 x} d x

Lösung

6 ln ∣ x ∣ - \frac{5}{2} ln x^{2} + 5 - \frac{1}{5} arctan (\frac{x}{5}) + C

Schritte zur Lösung

\int \frac{x ^{2} - x + 30}{x ^{3} + 5 x} d x

Ermittle den Partialbruch von

\frac{x ^{2} - x + 30}{x ^{3} + 5 x} : \frac{6}{x} + \frac{- 5 x - 1}{x ^{2} + 5}

= \int \frac{6}{x} + \frac{- 5 x - 1}{x ^{2} + 5} 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{6}{x} d x + \int \frac{- 5 x - 1}{x ^{2} + 5} d x

\int \frac{6}{x} d x = 6 ln ∣ x ∣

\int \frac{- 5 x - 1}{x ^{2} + 5} d x = - \frac{5}{2} ln x^{2} + 5 - \frac{1}{5} arctan (\frac{x}{5})

= 6 ln ∣ x ∣ - \frac{5}{2} ln x^{2} + 5 - \frac{1}{5} arctan (\frac{x}{5})

Füge eine Konstante zur Lösung hinzu

= 6 ln ∣ x ∣ - \frac{5}{2} ln x^{2} + 5 - \frac{1}{5} arctan (\frac{x}{5}) + C

Graph

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