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

Lösung

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

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

Schritte zur Lösung

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

Lange Division

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

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

\int x^{2} d x = \frac{x ^{3}}{3}

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

\int 1 d x = x

\int \frac{1}{- x + 1} d x = - ln ∣ - x + 1 ∣

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

Füge eine Konstante zur Lösung hinzu

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

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