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Beliebt Rechnen >

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

Lösung

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

Lösung

\frac{1}{3} x - \frac{x ^{2}}{2} + \frac{1}{3} x^{3} - \frac{x ^{4}}{12} + C

Schritte zur Lösung

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

Entferne die Konstante:

\int a \cdot f (x) d x = a \cdot \int f (x) d x

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

Multipliziere aus

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

= \frac{1}{3} \cdot \int 1 - 3 x + 3 x^{2} - 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

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

\int 1 d x = x

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

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

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

= \frac{1}{3} (x - \frac{3 x ^{2}}{2} + x^{3} - \frac{x ^{4}}{4})

Vereinfache

\frac{1}{3} (x - \frac{3 x ^{2}}{2} + x^{3} - \frac{x ^{4}}{4}) : \frac{1}{3} x - \frac{x ^{2}}{2} + \frac{1}{3} x^{3} - \frac{x ^{4}}{12}

= \frac{1}{3} x - \frac{x ^{2}}{2} + \frac{1}{3} x^{3} - \frac{x ^{4}}{12}

Füge eine Konstante zur Lösung hinzu

= \frac{1}{3} x - \frac{x ^{2}}{2} + \frac{1}{3} x^{3} - \frac{x ^{4}}{12} + C

Graph

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