integral from 4 to 12 of (4-x)(x-12)

解

\int_{4}^{12} (4 - x) (x - 12) d x

\frac{256}{3}

十進法表記

85.33333 \dots

解答ステップ

\int_{4}^{12} (4 - x) (x - 12) d x

拡張

(4 - x) (x - 12) : 16 x - x^{2} - 48

= \int_{4}^{12} 16 x - x^{2} - 48 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_{4}^{12} 16 x d x - \int_{4}^{12} x^{2} d x - \int_{4}^{12} 48 d x

\int_{4}^{12} 16 x d x = 1024

\int_{4}^{12} x^{2} d x = \frac{1664}{3}

\int_{4}^{12} 48 d x = 384

= 1024 - \frac{1664}{3} - 384

簡素化

1024 - \frac{1664}{3} - 384 : \frac{256}{3}

= \frac{256}{3}