integral from 0 to 1 of (1-x^4)^{11}

解

\int_{0}^{1} (1 - x^{4})^{11} d x

\frac{1073741824}{2187551925}

十進法表記

0.49084 \dots

解答ステップ

\int_{0}^{1} (1 - x^{4})^{11} d x

拡張

(1 - x^{4})^{11} : 1 - 11 x^{4} + 55 x^{8} - 165 x^{12} + 330 x^{16} - 462 x^{20} + 462 x^{24} - 330 x^{28} + 165 x^{32} - 55 x^{36} + 11 x^{40} - x^{44}

= \int_{0}^{1} 1 - 11 x^{4} + 55 x^{8} - 165 x^{12} + 330 x^{16} - 462 x^{20} + 462 x^{24} - 330 x^{28} + 165 x^{32} - 55 x^{36} + 11 x^{40} - x^{44} 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} 11 x^{4} d x + \int_{0}^{1} 55 x^{8} d x - \int_{0}^{1} 165 x^{12} d x + \int_{0}^{1} 330 x^{16} d x - \int_{0}^{1} 462 x^{20} d x + \int_{0}^{1} 462 x^{24} d x - \int_{0}^{1} 330 x^{28} d x + \int_{0}^{1} 165 x^{32} d x - \int_{0}^{1} 55 x^{36} d x + \int_{0}^{1} 11 x^{40} d x - \int_{0}^{1} x^{44} d x

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

\int_{0}^{1} 11 x^{4} d x = \frac{11}{5}

\int_{0}^{1} 55 x^{8} d x = \frac{55}{9}

\int_{0}^{1} 165 x^{12} d x = \frac{165}{13}

\int_{0}^{1} 330 x^{16} d x = \frac{330}{17}

\int_{0}^{1} 462 x^{20} d x = 22

\int_{0}^{1} 462 x^{24} d x = \frac{462}{25}

\int_{0}^{1} 330 x^{28} d x = \frac{330}{29}

\int_{0}^{1} 165 x^{32} d x = 5

\int_{0}^{1} 55 x^{36} d x = \frac{55}{37}

\int_{0}^{1} 11 x^{40} d x = \frac{11}{41}

\int_{0}^{1} x^{44} d x = \frac{1}{45}

= 1 - \frac{11}{5} + \frac{55}{9} - \frac{165}{13} + \frac{330}{17} - 22 + \frac{462}{25} - \frac{330}{29} + 5 - \frac{55}{37} + \frac{11}{41} - \frac{1}{45}

簡素化

1 - \frac{11}{5} + \frac{55}{9} - \frac{165}{13} + \frac{330}{17} - 22 + \frac{462}{25} - \frac{330}{29} + 5 - \frac{55}{37} + \frac{11}{41} - \frac{1}{45} : \frac{1073741824}{2187551925}

= \frac{1073741824}{2187551925}