integral from 1 to 6 of (1-x)(x-6)

解

\int_{1}^{6} (1 - x) (x - 6) d x

\frac{125}{6}

十進法表記

20.83333 \dots

解答ステップ

\int_{1}^{6} (1 - x) (x - 6) d x

拡張

(1 - x) (x - 6) : 7 x - x^{2} - 6

= \int_{1}^{6} 7 x - x^{2} - 6 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}^{6} 7 x d x - \int_{1}^{6} x^{2} d x - \int_{1}^{6} 6 d x

\int_{1}^{6} 7 x d x = \frac{245}{2}

\int_{1}^{6} x^{2} d x = \frac{215}{3}

\int_{1}^{6} 6 d x = 30

= \frac{245}{2} - \frac{215}{3} - 30

簡素化

\frac{245}{2} - \frac{215}{3} - 30 : \frac{125}{6}

= \frac{125}{6}