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

Lösung

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

\frac{x ^{8}}{8} - x^{7} + 3 x^{6} - 4 x^{5} + 2 x^{4} + C

Schritte zur Lösung

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

Multipliziere aus

(x - 1) (x^{2} - 2 x)^{3} : x^{7} - 7 x^{6} + 18 x^{5} - 20 x^{4} + 8 x^{3}

= \int x^{7} - 7 x^{6} + 18 x^{5} - 20 x^{4} + 8 x^{3} 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^{7} d x - \int 7 x^{6} d x + \int 18 x^{5} d x - \int 20 x^{4} d x + \int 8 x^{3} d x

\int x^{7} d x = \frac{x ^{8}}{8}

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

\int 18 x^{5} d x = 3 x^{6}

\int 20 x^{4} d x = 4 x^{5}

\int 8 x^{3} d x = 2 x^{4}

= \frac{x ^{8}}{8} - x^{7} + 3 x^{6} - 4 x^{5} + 2 x^{4}

Füge eine Konstante zur Lösung hinzu

= \frac{x ^{8}}{8} - x^{7} + 3 x^{6} - 4 x^{5} + 2 x^{4} + C

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