integral from 0 to 30 of 4x(30-x)

解

\int_{0}^{30} 4 x (30 - x) d x

18000

解答ステップ

\int_{0}^{30} 4 x (30 - x) d x

定数を除く:

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

= 4 \cdot \int_{0}^{30} x (30 - x) d x

拡張

x (30 - x) : 30 x - x^{2}

= 4 \cdot \int_{0}^{30} 30 x - 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

= 4 (\int_{0}^{30} 30 x d x - \int_{0}^{30} x^{2} d x)

\int_{0}^{30} 30 x d x = 13500

\int_{0}^{30} x^{2} d x = 9000

= 4 (13500 - 9000)

簡素化

= 18000