integral of x(x+1)^5

Lösung

\int x (x + 1)^{5} d x

\frac{x ^{7}}{7} + \frac{5 x ^{6}}{6} + 2 x^{5} + \frac{5 x ^{4}}{2} + \frac{5 x ^{3}}{3} + \frac{x ^{2}}{2} + C

Schritte zur Lösung

\int x (x + 1)^{5} d x

Multipliziere aus

x (x + 1)^{5} : x^{6} + 5 x^{5} + 10 x^{4} + 10 x^{3} + 5 x^{2} + x

= \int x^{6} + 5 x^{5} + 10 x^{4} + 10 x^{3} + 5 x^{2} + x 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^{6} d x + \int 5 x^{5} d x + \int 10 x^{4} d x + \int 10 x^{3} d x + \int 5 x^{2} d x + \int x d x

\int x^{6} d x = \frac{x ^{7}}{7}

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

\int 10 x^{4} d x = 2 x^{5}

\int 10 x^{3} d x = \frac{5 x ^{4}}{2}

\int 5 x^{2} d x = \frac{5 x ^{3}}{3}

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

= \frac{x ^{7}}{7} + \frac{5 x ^{6}}{6} + 2 x^{5} + \frac{5 x ^{4}}{2} + \frac{5 x ^{3}}{3} + \frac{x ^{2}}{2}

Füge eine Konstante zur Lösung hinzu

= \frac{x ^{7}}{7} + \frac{5 x ^{6}}{6} + 2 x^{5} + \frac{5 x ^{4}}{2} + \frac{5 x ^{3}}{3} + \frac{x ^{2}}{2} + C

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