integral from 1 to 2 of x(x-1)^9

解

\int_{1}^{2} x (x - 1)^{9} d x

\frac{21}{110}

十進法表記

0.19090 \dots

解答ステップ

\int_{1}^{2} x (x - 1)^{9} d x

拡張

x (x - 1)^{9} : x^{10} - 9 x^{9} + 36 x^{8} - 84 x^{7} + 126 x^{6} - 126 x^{5} + 84 x^{4} - 36 x^{3} + 9 x^{2} - x

= \int_{1}^{2} x^{10} - 9 x^{9} + 36 x^{8} - 84 x^{7} + 126 x^{6} - 126 x^{5} + 84 x^{4} - 36 x^{3} + 9 x^{2} - x 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_{1}^{2} x^{10} d x - \int_{1}^{2} 9 x^{9} d x + \int_{1}^{2} 36 x^{8} d x - \int_{1}^{2} 84 x^{7} d x + \int_{1}^{2} 126 x^{6} d x - \int_{1}^{2} 126 x^{5} d x + \int_{1}^{2} 84 x^{4} d x - \int_{1}^{2} 36 x^{3} d x + \int_{1}^{2} 9 x^{2} d x - \int_{1}^{2} x d x

\int_{1}^{2} x^{10} d x = \frac{2047}{11}

\int_{1}^{2} 9 x^{9} d x = \frac{9207}{10}

\int_{1}^{2} 36 x^{8} d x = 2044

\int_{1}^{2} 84 x^{7} d x = \frac{5355}{2}

\int_{1}^{2} 126 x^{6} d x = 2286

\int_{1}^{2} 126 x^{5} d x = 1323

\int_{1}^{2} 84 x^{4} d x = \frac{2604}{5}

\int_{1}^{2} 36 x^{3} d x = 135

\int_{1}^{2} 9 x^{2} d x = 21

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

= \frac{2047}{11} - \frac{9207}{10} + 2044 - \frac{5355}{2} + 2286 - 1323 + \frac{2604}{5} - 135 + 21 - \frac{3}{2}

簡素化

\frac{2047}{11} - \frac{9207}{10} + 2044 - \frac{5355}{2} + 2286 - 1323 + \frac{2604}{5} - 135 + 21 - \frac{3}{2} : \frac{21}{110}

= \frac{21}{110}