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Beliebt Rechnen >

integral of (x^3)/(x^3-1)

Lösung

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

Lösung

x + \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{x ^{3}}{x ^{3} - 1} d x

Ermittle den Partialbruch von

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

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

\int 1 d x = 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}))

= x + \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

x + \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})) : x + \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))

= x + \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

= x + \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

f(x)=(\sqrt[3]{x^2-9})/x

f (x) = \frac{3 x ^{2} - 9}{x}

integral of 2xy^2+3x^2

\int 2 x y^{2} + 3 x^{2} d x

(\partial)/(\partial y)(y^2sin(x/y))

\frac{\partial}{\partial y} (y^{2} sin (\frac{x}{y}))

fläche y=x^3,y=x^2-2x,[-1,1]

a re a y = x^{3}, y = x^{2} - 2 x, [- 1, 1]

derivative 2/3 x^3+x^2-12x+12

d er i v a t i v e \frac{2}{3} x^{3} + x^{2} - 12 x + 12