integral of x(1-x)^3

Lösung

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

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

Schritte zur Lösung

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

Multipliziere aus

x (1 - x)^{3} : x - 3 x^{2} + 3 x^{3} - x^{4}

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

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

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

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

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

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

Füge eine Konstante zur Lösung hinzu

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

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