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Beliebt Rechnen >

integral of (x^3+4)/(x^5+14x^3+40x)

Lösung

\int \frac{x ^{3} + 4}{x ^{5} + 14 x ^{3} + 40 x} d x

Lösung

\frac{1}{10} ln ∣ x ∣ + \frac{1}{6} (- \frac{1}{2} ln x^{2} + 4 - 2 arctan (\frac{x}{2})) + \frac{1}{15} (\frac{1}{2} ln x^{2} + 10 + \frac{5 5}{2} arctan (\frac{x}{10})) + C

Schritte zur Lösung

\int \frac{x ^{3} + 4}{x ^{5} + 14 x ^{3} + 40 x} d x

Ermittle den Partialbruch von

\frac{x ^{3} + 4}{x ^{5} + 14 x ^{3} + 40 x} : \frac{1}{10 x} + \frac{- x - 4}{6 ( x ^{2} + 4 )} + \frac{x + 25}{15 ( x ^{2} + 10 )}

= \int \frac{1}{10 x} + \frac{- x - 4}{6 ( x ^{2} + 4 )} + \frac{x + 25}{15 ( x ^{2} + 10 )} 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}{10 x} d x + \int \frac{- x - 4}{6 ( x ^{2} + 4 )} d x + \int \frac{x + 25}{15 ( x ^{2} + 10 )} d x

\int \frac{1}{10 x} d x = \frac{1}{10} ln ∣ x ∣

\int \frac{- x - 4}{6 ( x ^{2} + 4 )} d x = \frac{1}{6} (- \frac{1}{2} ln x^{2} + 4 - 2 arctan (\frac{x}{2}))

\int \frac{x + 25}{15 ( x ^{2} + 10 )} d x = \frac{1}{15} (\frac{1}{2} ln x^{2} + 10 + \frac{5 5}{2} arctan (\frac{x}{10}))

= \frac{1}{10} ln ∣ x ∣ + \frac{1}{6} (- \frac{1}{2} ln x^{2} + 4 - 2 arctan (\frac{x}{2})) + \frac{1}{15} (\frac{1}{2} ln x^{2} + 10 + \frac{5 5}{2} arctan (\frac{x}{10}))

Füge eine Konstante zur Lösung hinzu

= \frac{1}{10} ln ∣ x ∣ + \frac{1}{6} (- \frac{1}{2} ln x^{2} + 4 - 2 arctan (\frac{x}{2})) + \frac{1}{15} (\frac{1}{2} ln x^{2} + 10 + \frac{5 5}{2} arctan (\frac{x}{10})) + C

Graph

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