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Beliebt Rechnen >

integral of 1/(x^3-8)

Lösung

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

Lösung

\frac{1}{12} ln ∣ x - 2 ∣ - \frac{1}{24} ln x^{2} + 2 x + 4 - \frac{1}{4 3} arctan (\frac{1}{3} (x + 1)) + C

Schritte zur Lösung

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

Ermittle den Partialbruch von

\frac{1}{x ^{3} - 8} : \frac{1}{12 ( x - 2 )} + \frac{- x - 4}{12 ( x ^{2} + 2 x + 4 )}

= \int \frac{1}{12 ( x - 2 )} + \frac{- x - 4}{12 ( x ^{2} + 2 x + 4 )} 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}{12 ( x - 2 )} d x + \int \frac{- x - 4}{12 ( x ^{2} + 2 x + 4 )} d x

\int \frac{1}{12 ( x - 2 )} d x = \frac{1}{12} ln ∣ x - 2 ∣

\int \frac{- x - 4}{12 ( x ^{2} + 2 x + 4 )} d x = \frac{1}{24 3} (- 3 ln x^{2} + 2 x + 4 - 6 arctan (\frac{1}{3} (x + 1)))

= \frac{1}{12} ln ∣ x - 2 ∣ + \frac{1}{24 3} (- 3 ln x^{2} + 2 x + 4 - 6 arctan (\frac{1}{3} (x + 1)))

Multipliziere aus

\frac{1}{24 3} (- 3 ln x^{2} + 2 x + 4 - 6 arctan (\frac{1}{3} (x + 1))) : - \frac{1}{24} ln x^{2} + 2 x + 4 - \frac{1}{4 3} arctan (\frac{1}{3} (x + 1))

= \frac{1}{12} ln ∣ x - 2 ∣ - \frac{1}{24} ln x^{2} + 2 x + 4 - \frac{1}{4 3} arctan (\frac{1}{3} (x + 1))

Füge eine Konstante zur Lösung hinzu

= \frac{1}{12} ln ∣ x - 2 ∣ - \frac{1}{24} ln x^{2} + 2 x + 4 - \frac{1}{4 3} arctan (\frac{1}{3} (x + 1)) + C

Graph

Beliebte Beispiele

(\partial)/(\partial z)(x-z-arctan(yz))

\frac{\partial}{\partial z} (x - z - arctan (yz))

integral of 6xsec(x)tan(x)

\int 6 x sec (x) tan (x) d x

derivative of ln(x^2+10)

\frac{d}{d x} (ln (x^{2} + 10))

d/(dt)(10000(12+t)e^{-0.05t})

\frac{d}{d t} (10000 (12 + t) e^{- 0.05 t})

integral of 5/((3x-1))

\int \frac{5}{( 3 x - 1 )} d x