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Beliebt Rechnen >

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

Lösung

\int \frac{( x ^{4} - x ^{3} - x - 1 )}{x ^{3} - x ^{2}} d x

Lösung

\frac{x ^{2}}{2} + 2 ln ∣ x ∣ - \frac{1}{x} - 2 ln ∣ x - 1 ∣ + C

Schritte zur Lösung

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

Lange Division

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

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

\int x d x = \frac{x ^{2}}{2}

\int \frac{- x - 1}{x ^{3} - x ^{2}} d x = 2 ln ∣ x ∣ - \frac{1}{x} - 2 ln ∣ x - 1 ∣

= \frac{x ^{2}}{2} + 2 ln ∣ x ∣ - \frac{1}{x} - 2 ln ∣ x - 1 ∣

Füge eine Konstante zur Lösung hinzu

= \frac{x ^{2}}{2} + 2 ln ∣ x ∣ - \frac{1}{x} - 2 ln ∣ x - 1 ∣ + C

Graph

Beliebte Beispiele

integral of 85-3sqrt(25-x)

\int 85 - 3 25 - x d x

derivative of 6x^{-2}+9x^{-1}

\frac{d}{d x} (6 x^{- 2} + 9 x^{- 1})

derivative of 2sqrt(x)-x

\frac{d}{d x} (2 x - x)

sum from n=1 to infinity of (n^2)/(e^n)

n = 1 \sum \infty \frac{n ^{2}}{e ^{n}}

limit as x approaches infinity of sqrt(x^2-9),+x

x \to \infty lim (x^{2} - 9, + x)