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

解

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

\frac{1}{15}

十進法表記

0.06666 \dots

解答ステップ

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

拡張

x (x - x) : x^{\frac{3}{2}} - x^{2}

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

\int_{0}^{1} x^{\frac{3}{2}} d x = \frac{2}{5}

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

= \frac{2}{5} - \frac{1}{3}

簡素化

\frac{2}{5} - \frac{1}{3} : \frac{1}{15}

= \frac{1}{15}