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

Lösung

\int_{0}^{2} \frac{3}{2} x^{2} (x - 1)^{2} d x

\frac{8}{5}

Dezimale

1.6

Schritte zur Lösung

\int_{0}^{2} \frac{3}{2} x^{2} (x - 1)^{2} d x

Entferne die Konstante:

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

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

Multipliziere aus

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

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

\int_{0}^{2} x^{4} d x = \frac{32}{5}

\int_{0}^{2} 2 x^{3} d x = 8

\int_{0}^{2} x^{2} d x = \frac{8}{3}

= \frac{3}{2} (\frac{32}{5} - 8 + \frac{8}{3})

Vereinfache

\frac{3}{2} (\frac{32}{5} - 8 + \frac{8}{3}) : \frac{8}{5}

= \frac{8}{5}

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