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

解

\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}

解答ステップ

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

拡張

(- 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

総和規則を適用する:

\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}