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Beliebt Rechnen >

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

Lösung

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

Lösung

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

Schritte zur Lösung

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

Wende U-Substitution an

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

Multipliziere aus

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

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

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

\int \frac{1}{u} d u = ln ∣ u ∣

\int \frac{( u - 1 ) ^{3}}{u ^{2}} d u = \frac{u ^{2}}{2} - 3 u + 3 ln ∣ u ∣ + \frac{1}{u}

= - ln ∣ u ∣ + \frac{u ^{2}}{2} - 3 u + 3 ln ∣ u ∣ + \frac{1}{u}

Setze in

u = x + 1

ein

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

Vereinfache

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

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

Füge eine Konstante zur Lösung hinzu

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

Graph

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