integral from 0 to 1 of (1-x^2)^5

解

\int_{0}^{1} (1 - x^{2})^{5} d x

\frac{256}{693}

十進法表記

0.36940 \dots

解答ステップ

\int_{0}^{1} (1 - x^{2})^{5} d x

拡張

(1 - x^{2})^{5} : 1 - 5 x^{2} + 10 x^{4} - 10 x^{6} + 5 x^{8} - x^{10}

= \int_{0}^{1} 1 - 5 x^{2} + 10 x^{4} - 10 x^{6} + 5 x^{8} - x^{10} 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} 5 x^{2} d x + \int_{0}^{1} 10 x^{4} d x - \int_{0}^{1} 10 x^{6} d x + \int_{0}^{1} 5 x^{8} d x - \int_{0}^{1} x^{10} d x

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

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

\int_{0}^{1} 10 x^{4} d x = 2

\int_{0}^{1} 10 x^{6} d x = \frac{10}{7}

\int_{0}^{1} 5 x^{8} d x = \frac{5}{9}

\int_{0}^{1} x^{10} d x = \frac{1}{11}

= 1 - \frac{5}{3} + 2 - \frac{10}{7} + \frac{5}{9} - \frac{1}{11}

簡素化

1 - \frac{5}{3} + 2 - \frac{10}{7} + \frac{5}{9} - \frac{1}{11} : \frac{256}{693}

= \frac{256}{693}