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Beliebt Rechnen >

integral of 1/(x^3+27)

Lösung

\int \frac{1}{x ^{3} + 27} d x

Lösung

\frac{1}{27} ln ∣ x + 3 ∣ + \frac{1}{54} (- ln 4 x^{2} - 12 x + 36 + 23 arctan (\frac{1}{3 3} (2 x - 3))) + C

Schritte zur Lösung

\int \frac{1}{x ^{3} + 27} d x

Ermittle den Partialbruch von

\frac{1}{x ^{3} + 27} : \frac{1}{27 ( x + 3 )} + \frac{- x + 6}{27 ( x ^{2} - 3 x + 9 )}

= \int \frac{1}{27 ( x + 3 )} + \frac{- x + 6}{27 ( x ^{2} - 3 x + 9 )} 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}{27 ( x + 3 )} d x + \int \frac{- x + 6}{27 ( x ^{2} - 3 x + 9 )} d x

\int \frac{1}{27 ( x + 3 )} d x = \frac{1}{27} ln ∣ x + 3 ∣

\int \frac{- x + 6}{27 ( x ^{2} - 3 x + 9 )} d x = \frac{1}{54} (- ln 4 x^{2} - 12 x + 36 + 23 arctan (\frac{2 x - 3 3}{3 3}))

= \frac{1}{27} ln ∣ x + 3 ∣ + \frac{1}{54} (- ln 4 x^{2} - 12 x + 36 + 23 arctan (\frac{2 x - 3 3}{3 3}))

Vereinfache

\frac{1}{27} ln ∣ x + 3 ∣ + \frac{1}{54} (- ln 4 x^{2} - 12 x + 36 + 23 arctan (\frac{2 x - 3 3}{3 3})) : \frac{1}{27} ln ∣ x + 3 ∣ + \frac{1}{54} (- ln 4 x^{2} - 12 x + 36 + 23 arctan (\frac{1}{3 3} (2 x - 3)))

= \frac{1}{27} ln ∣ x + 3 ∣ + \frac{1}{54} (- ln 4 x^{2} - 12 x + 36 + 23 arctan (\frac{1}{3 3} (2 x - 3)))

Füge eine Konstante zur Lösung hinzu

= \frac{1}{27} ln ∣ x + 3 ∣ + \frac{1}{54} (- ln 4 x^{2} - 12 x + 36 + 23 arctan (\frac{1}{3 3} (2 x - 3))) + C

Graph

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