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

解答

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

\frac{8191}{26}

十进制

315.03846 \dots

求解步骤

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

乘开

x (2 - x^{2})^{12} : 4096 x - 24576 x^{3} + 67584 x^{5} - 112640 x^{7} + 126720 x^{9} - 101376 x^{11} + 59136 x^{13} - 25344 x^{15} + 7920 x^{17} - 1760 x^{19} + 264 x^{21} - 24 x^{23} + x^{25}

= \int_{0}^{1} 4096 x - 24576 x^{3} + 67584 x^{5} - 112640 x^{7} + 126720 x^{9} - 101376 x^{11} + 59136 x^{13} - 25344 x^{15} + 7920 x^{17} - 1760 x^{19} + 264 x^{21} - 24 x^{23} + x^{25} d x

使用积分加法定则:

\int f (x) \pm g (x) d x = \int f (x) d x \pm \int g (x) d x

= \int_{0}^{1} 4096 x d x - \int_{0}^{1} 24576 x^{3} d x + \int_{0}^{1} 67584 x^{5} d x - \int_{0}^{1} 112640 x^{7} d x + \int_{0}^{1} 126720 x^{9} d x - \int_{0}^{1} 101376 x^{11} d x + \int_{0}^{1} 59136 x^{13} d x - \int_{0}^{1} 25344 x^{15} d x + \int_{0}^{1} 7920 x^{17} d x - \int_{0}^{1} 1760 x^{19} d x + \int_{0}^{1} 264 x^{21} d x - \int_{0}^{1} 24 x^{23} d x + \int_{0}^{1} x^{25} d x

\int_{0}^{1} 4096 x d x = 2048

\int_{0}^{1} 24576 x^{3} d x = 6144

\int_{0}^{1} 67584 x^{5} d x = 11264

\int_{0}^{1} 112640 x^{7} d x = 14080

\int_{0}^{1} 126720 x^{9} d x = 12672

\int_{0}^{1} 101376 x^{11} d x = 8448

\int_{0}^{1} 59136 x^{13} d x = 4224

\int_{0}^{1} 25344 x^{15} d x = 1584

\int_{0}^{1} 7920 x^{17} d x = 440

\int_{0}^{1} 1760 x^{19} d x = 88

\int_{0}^{1} 264 x^{21} d x = 12

\int_{0}^{1} 24 x^{23} d x = 1

\int_{0}^{1} x^{25} d x = \frac{1}{26}

= 2048 - 6144 + 11264 - 14080 + 12672 - 8448 + 4224 - 1584 + 440 - 88 + 12 - 1 + \frac{1}{26}

化简

2048 - 6144 + 11264 - 14080 + 12672 - 8448 + 4224 - 1584 + 440 - 88 + 12 - 1 + \frac{1}{26} : \frac{8191}{26}

= \frac{8191}{26}

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