integral from 0 to 1 of (x-x^2)x

解

\int_{0}^{1} (x - x^{2}) x d x

\frac{1}{12}

十進法表記

0.08333 \dots

解答ステップ

\int_{0}^{1} (x - x^{2}) x d x

拡張

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

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

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

\int_{0}^{1} x^{3} d x = \frac{1}{4}

= \frac{1}{3} - \frac{1}{4}

簡素化

\frac{1}{3} - \frac{1}{4} : \frac{1}{12}

= \frac{1}{12}