integral from 1 to 5 of x(x^2-1)^2

解

\int_{1}^{5} x (x^{2} - 1)^{2} d x

2304

解答ステップ

\int_{1}^{5} x (x^{2} - 1)^{2} d x

拡張

x (x^{2} - 1)^{2} : x^{5} - 2 x^{3} + x

= \int_{1}^{5} x^{5} - 2 x^{3} + x 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_{1}^{5} x^{5} d x - \int_{1}^{5} 2 x^{3} d x + \int_{1}^{5} x d x

\int_{1}^{5} x^{5} d x = 2604

\int_{1}^{5} 2 x^{3} d x = 312

\int_{1}^{5} x d x = 12

= 2604 - 312 + 12

簡素化

= 2304