integral from 1 to 10 of 2/81 x(x-1)

解

\int_{1}^{10} \frac{2}{81} x (x - 1) d x

7

解答ステップ

\int_{1}^{10} \frac{2}{81} x (x - 1) d x

定数を除く:

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

= \frac{2}{81} \cdot \int_{1}^{10} x (x - 1) d x

拡張

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

= \frac{2}{81} \cdot \int_{1}^{10} x^{2} - x 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

= \frac{2}{81} (\int_{1}^{10} x^{2} d x - \int_{1}^{10} x d x)

\int_{1}^{10} x^{2} d x = 333

\int_{1}^{10} x d x = \frac{99}{2}

= \frac{2}{81} (333 - \frac{99}{2})

簡素化

\frac{2}{81} (333 - \frac{99}{2}) : 7

= 7