integral from 0 to 1 of ((x-sqrt(x))/3)

解

\int_{0}^{1} (\frac{x - x}{3}) d x

- \frac{1}{18}

十進法表記

- 0.05555 \dots

解答ステップ

\int_{0}^{1} \frac{x - x}{3} d x

定数を除く:

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

= \frac{1}{3} \cdot \int_{0}^{1} x - 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{1}{3} (\int_{0}^{1} x d x - \int_{0}^{1} x d x)

\int_{0}^{1} x d x = \frac{1}{2}

\int_{0}^{1} x d x = \frac{2}{3}

= \frac{1}{3} (\frac{1}{2} - \frac{2}{3})

簡素化

\frac{1}{3} (\frac{1}{2} - \frac{2}{3}) : - \frac{1}{18}

= - \frac{1}{18}