integral from 0 to 2 of x(x^2-1)^3

解

\int_{0}^{2} x (x^{2} - 1)^{3} d x

10

解答ステップ

\int_{0}^{2} x (x^{2} - 1)^{3} d x

拡張

x (x^{2} - 1)^{3} : x^{7} - 3 x^{5} + 3 x^{3} - x

= \int_{0}^{2} x^{7} - 3 x^{5} + 3 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_{0}^{2} x^{7} d x - \int_{0}^{2} 3 x^{5} d x + \int_{0}^{2} 3 x^{3} d x - \int_{0}^{2} x d x

\int_{0}^{2} x^{7} d x = 32

\int_{0}^{2} 3 x^{5} d x = 32

\int_{0}^{2} 3 x^{3} d x = 12

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

= 32 - 32 + 12 - 2

簡素化

= 10