integral from-1 to 5 of 4x(1-x^2)

解

\int_{- 1}^{5} 4 x (1 - x^{2}) d x

- 576

解答ステップ

\int_{- 1}^{5} 4 x (1 - x^{2}) d x

定数を除く:

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

= 4 \cdot \int_{- 1}^{5} x (1 - x^{2}) d x

拡張

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

= 4 \cdot \int_{- 1}^{5} x - x^{3} 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

= 4 (\int_{- 1}^{5} x d x - \int_{- 1}^{5} x^{3} d x)

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

\int_{- 1}^{5} x^{3} d x = 156

= 4 (12 - 156)

簡素化

= - 576