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Beliebt Rechnen >

integral from 0 to 1 of pi(x-x^3)^2

Lösung

\int_{0}^{1} π (x - x^{3})^{2} d x

Lösung

π \frac{8}{105}

+1

Dezimale

0.23935 \dots

Schritte zur Lösung

\int_{0}^{1} π (x - x^{3})^{2} d x

Entferne die Konstante:

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

= π \cdot \int_{0}^{1} (x - x^{3})^{2} d x

Multipliziere aus

(x - x^{3})^{2} : x^{2} - 2 x^{4} + x^{6}

= π \cdot \int_{0}^{1} x^{2} - 2 x^{4} + x^{6} 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

= π (\int_{0}^{1} x^{2} d x - \int_{0}^{1} 2 x^{4} d x + \int_{0}^{1} x^{6} d x)

\int_{0}^{1} x^{2} d x = \frac{1}{3}

\int_{0}^{1} 2 x^{4} d x = \frac{2}{5}

\int_{0}^{1} x^{6} d x = \frac{1}{7}

= π (\frac{1}{3} - \frac{2}{5} + \frac{1}{7})

Füge

\frac{1}{3} - \frac{2}{5} + \frac{1}{7}

zusammen:

\frac{8}{105}

= π \frac{8}{105}

Graph

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