integral from 1 to 7 of (1-x)(x-7)

解

\int_{1}^{7} (1 - x) (x - 7) d x

36

解答ステップ

\int_{1}^{7} (1 - x) (x - 7) d x

拡張

(1 - x) (x - 7) : 8 x - x^{2} - 7

= \int_{1}^{7} 8 x - x^{2} - 7 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}^{7} 8 x d x - \int_{1}^{7} x^{2} d x - \int_{1}^{7} 7 d x

\int_{1}^{7} 8 x d x = 192

\int_{1}^{7} x^{2} d x = 114

\int_{1}^{7} 7 d x = 42

= 192 - 114 - 42

簡素化

= 36