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Beliebt Rechnen >

integral of 1/(x^3-1)

Lösung

\int \frac{1}{x ^{3} - 1} d x

Lösung

\frac{1}{3} ln ∣ x - 1 ∣ - \frac{1}{6} ln 4 x^{2} + 4 x + 4 - \frac{1}{3} arctan (\frac{1}{3} (2 x + 1)) + C

Schritte zur Lösung

\int \frac{1}{x ^{3} - 1} d x

Ermittle den Partialbruch von

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

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

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

\int \frac{- x - 2}{3 ( x ^{2} + x + 1 )} d x = \frac{1}{6 3} (- 3 ln 4 x^{2} + 4 x + 4 - 6 arctan (\frac{2 x + 1}{3}))

= \frac{1}{3} ln ∣ x - 1 ∣ + \frac{1}{6 3} (- 3 ln 4 x^{2} + 4 x + 4 - 6 arctan (\frac{2 x + 1}{3}))

Vereinfache

\frac{1}{3} ln ∣ x - 1 ∣ + \frac{1}{6 3} (- 3 ln 4 x^{2} + 4 x + 4 - 6 arctan (\frac{2 x + 1}{3})) : \frac{1}{3} ln ∣ x - 1 ∣ - \frac{1}{6} ln 4 x^{2} + 4 x + 4 - \frac{1}{3} arctan (\frac{1}{3} (2 x + 1))

= \frac{1}{3} ln ∣ x - 1 ∣ - \frac{1}{6} ln 4 x^{2} + 4 x + 4 - \frac{1}{3} arctan (\frac{1}{3} (2 x + 1))

Füge eine Konstante zur Lösung hinzu

= \frac{1}{3} ln ∣ x - 1 ∣ - \frac{1}{6} ln 4 x^{2} + 4 x + 4 - \frac{1}{3} arctan (\frac{1}{3} (2 x + 1)) + C

Graph

Beliebte Beispiele

implicit (dy)/(dx),x^2-y^3=6

im pl i c i t \frac{d y}{d x}, x^{2} - y^{3} = 6

integral of cos(x)+1/3 sin(x)-1/(x^2)

\int cos (x) + \frac{1}{3} sin (x) - \frac{1}{x ^{2}} d x

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\frac{d}{d x} ((2^{x + 1} - 3. 5^{x})^{3})

integral of 5x+sin(x)

\int 5 x + sin (x) d x

d er i v a t i v e \frac{1}{c}