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

Lösung

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

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

Schritte zur Lösung

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

Wende U-Substitution an

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

Multipliziere aus

\frac{u ^{2} - 3 u + 2}{u} : u - 3 + \frac{2}{u}

= \int u - 3 + \frac{2}{u} 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 u d u - \int 3 d u + \int \frac{2}{u} d u

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

\int 3 d u = 3 u

\int \frac{2}{u} d u = 2 ln ∣ u ∣

= \frac{u ^{2}}{2} - 3 u + 2 ln ∣ u ∣

Setze in

u = x + 1

ein

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

Füge eine Konstante zur Lösung hinzu

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

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