integral from 0 to 1 of (1-x)^3

解

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

\frac{1}{4}

十進法表記

0.25

解答ステップ

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

拡張

(1 - x)^{3} : 1 - 3 x + 3 x^{2} - x^{3}

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

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

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

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

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

= 1 - \frac{3}{2} + 1 - \frac{1}{4}

簡素化

1 - \frac{3}{2} + 1 - \frac{1}{4} : \frac{1}{4}

= \frac{1}{4}