integral of (x^6)/(1+x)

Lösung

\int \frac{x ^{6}}{1 + x} d x

\frac{x ^{6}}{6} - \frac{x ^{5}}{5} + \frac{x ^{4}}{4} - \frac{x ^{3}}{3} + \frac{x ^{2}}{2} - x + ln ∣ x + 1 ∣ + C

Schritte zur Lösung

\int \frac{x ^{6}}{1 + x} d x

Lange Division

\frac{x ^{6}}{1 + x} : x^{5} - x^{4} + x^{3} - x^{2} + x - 1 + \frac{1}{x + 1}

= \int x^{5} - x^{4} + x^{3} - 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^{5} d x - \int x^{4} d x + \int x^{3} d x - \int x^{2} d x + \int x d x - \int 1 d x + \int \frac{1}{x + 1} d x

\int x^{5} d x = \frac{x ^{6}}{6}

\int x^{4} d x = \frac{x ^{5}}{5}

\int x^{3} d x = \frac{x ^{4}}{4}

\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 ^{6}}{6} - \frac{x ^{5}}{5} + \frac{x ^{4}}{4} - \frac{x ^{3}}{3} + \frac{x ^{2}}{2} - x + ln ∣ x + 1 ∣

Füge eine Konstante zur Lösung hinzu

= \frac{x ^{6}}{6} - \frac{x ^{5}}{5} + \frac{x ^{4}}{4} - \frac{x ^{3}}{3} + \frac{x ^{2}}{2} - x + ln ∣ x + 1 ∣ + C