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Beliebt Rechnen >

integral of 1/(x^3-4x^2+8x)

Lösung

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

Lösung

\frac{1}{8} ln ∣ x ∣ + \frac{1}{16} (- ln x^{2} - 4 x + 8 + 2 arctan (\frac{x - 2}{2})) + C

Schritte zur Lösung

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

Ermittle den Partialbruch von

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

= \int \frac{1}{8 x} + \frac{- x + 4}{8 ( x ^{2} - 4 x + 8 )} 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}{8 x} d x + \int \frac{- x + 4}{8 ( x ^{2} - 4 x + 8 )} d x

\int \frac{1}{8 x} d x = \frac{1}{8} ln ∣ x ∣

\int \frac{- x + 4}{8 ( x ^{2} - 4 x + 8 )} d x = \frac{1}{16} (- ln x^{2} - 4 x + 8 + 2 arctan (\frac{x - 2}{2}))

= \frac{1}{8} ln ∣ x ∣ + \frac{1}{16} (- ln x^{2} - 4 x + 8 + 2 arctan (\frac{x - 2}{2}))

Füge eine Konstante zur Lösung hinzu

= \frac{1}{8} ln ∣ x ∣ + \frac{1}{16} (- ln x^{2} - 4 x + 8 + 2 arctan (\frac{x - 2}{2})) + C

Graph

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