integral from 0 to 1 of (1-x^2)^{12}

解

\int_{0}^{1} (1 - x^{2})^{12} d x

\frac{4194304}{16900975}

十進法表記

0.24816 \dots

解答ステップ

\int_{0}^{1} (1 - x^{2})^{12} d x

拡張

(1 - x^{2})^{12} : 1 - 12 x^{2} + 66 x^{4} - 220 x^{6} + 495 x^{8} - 792 x^{10} + 924 x^{12} - 792 x^{14} + 495 x^{16} - 220 x^{18} + 66 x^{20} - 12 x^{22} + x^{24}

= \int_{0}^{1} 1 - 12 x^{2} + 66 x^{4} - 220 x^{6} + 495 x^{8} - 792 x^{10} + 924 x^{12} - 792 x^{14} + 495 x^{16} - 220 x^{18} + 66 x^{20} - 12 x^{22} + x^{24} 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} 12 x^{2} d x + \int_{0}^{1} 66 x^{4} d x - \int_{0}^{1} 220 x^{6} d x + \int_{0}^{1} 495 x^{8} d x - \int_{0}^{1} 792 x^{10} d x + \int_{0}^{1} 924 x^{12} d x - \int_{0}^{1} 792 x^{14} d x + \int_{0}^{1} 495 x^{16} d x - \int_{0}^{1} 220 x^{18} d x + \int_{0}^{1} 66 x^{20} d x - \int_{0}^{1} 12 x^{22} d x + \int_{0}^{1} x^{24} d x

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

\int_{0}^{1} 12 x^{2} d x = 4

\int_{0}^{1} 66 x^{4} d x = \frac{66}{5}

\int_{0}^{1} 220 x^{6} d x = \frac{220}{7}

\int_{0}^{1} 495 x^{8} d x = 55

\int_{0}^{1} 792 x^{10} d x = 72

\int_{0}^{1} 924 x^{12} d x = \frac{924}{13}

\int_{0}^{1} 792 x^{14} d x = \frac{264}{5}

\int_{0}^{1} 495 x^{16} d x = \frac{495}{17}

\int_{0}^{1} 220 x^{18} d x = \frac{220}{19}

\int_{0}^{1} 66 x^{20} d x = \frac{22}{7}

\int_{0}^{1} 12 x^{22} d x = \frac{12}{23}

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

= 1 - 4 + \frac{66}{5} - \frac{220}{7} + 55 - 72 + \frac{924}{13} - \frac{264}{5} + \frac{495}{17} - \frac{220}{19} + \frac{22}{7} - \frac{12}{23} + \frac{1}{25}

簡素化

1 - 4 + \frac{66}{5} - \frac{220}{7} + 55 - 72 + \frac{924}{13} - \frac{264}{5} + \frac{495}{17} - \frac{220}{19} + \frac{22}{7} - \frac{12}{23} + \frac{1}{25} : \frac{4194304}{16900975}

= \frac{4194304}{16900975}