integral from 0 to 12 of 1/36 (x(12-x))

Lösung

\int_{0}^{12} \frac{1}{36} (x (12 - x)) d x

8

Schritte zur Lösung

\int_{0}^{12} \frac{1}{36} x (12 - x) d x

Entferne die Konstante:

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

= \frac{1}{36} \cdot \int_{0}^{12} x (12 - x) d x

Multipliziere aus

x (12 - x) : 12 x - x^{2}

= \frac{1}{36} \cdot \int_{0}^{12} 12 x - 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{1}{36} (\int_{0}^{12} 12 x d x - \int_{0}^{12} x^{2} d x)

\int_{0}^{12} 12 x d x = 864

\int_{0}^{12} x^{2} d x = 576

= \frac{1}{36} (864 - 576)

Vereinfache

\frac{1}{36} (864 - 576) : 8

= 8

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