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

Lösung

\int_{0}^{1} k x^{2} (1 - x)^{2} d x

\frac{1}{30} k

Schritte zur Lösung

\int_{0}^{1} k x^{2} (1 - x)^{2} d x

Entferne die Konstante:

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

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

Multipliziere aus

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

= k \cdot \int_{0}^{1} x^{2} - 2 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

= k (\int_{0}^{1} x^{2} d x - \int_{0}^{1} 2 x^{3} d x + \int_{0}^{1} x^{4} d x)

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

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

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

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

Füge

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

zusammen:

\frac{1}{30}

= \frac{1}{30} k

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