integral from 0 to L of x^2(L-x)(2L-x)

Lösung

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

- \frac{3 L ^{5}}{4} + \frac{L ^{5}}{5} + \frac{2 L ^{5}}{3}

Schritte zur Lösung

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

Multipliziere aus

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

= \int_{0}^{L} - 3 L x^{3} + x^{4} + 2 L^{2} 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

= - \int_{0}^{L} 3 L x^{3} d x + \int_{0}^{L} x^{4} d x + \int_{0}^{L} 2 L^{2} x^{2} d x

\int_{0}^{L} 3 L x^{3} d x = \frac{3 L ^{5}}{4}

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

\int_{0}^{L} 2 L^{2} x^{2} d x = \frac{2 L ^{5}}{3}

= - \frac{3 L ^{5}}{4} + \frac{L ^{5}}{5} + \frac{2 L ^{5}}{3}

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