integral from 0 to 4 of 3x(4-x)

Lösung

\int_{0}^{4} 3 x (4 - x) d x

32

Schritte zur Lösung

\int_{0}^{4} 3 x (4 - x) d x

Entferne die Konstante:

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

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

Multipliziere aus

x (4 - x) : 4 x - x^{2}

= 3 \cdot \int_{0}^{4} 4 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

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

\int_{0}^{4} 4 x d x = 32

\int_{0}^{4} x^{2} d x = \frac{64}{3}

= 3 (32 - \frac{64}{3})

Vereinfache

3 (32 - \frac{64}{3}) : 32

= 32

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