integral from 0 to 1 of x^2(x+1)^6

解

\int_{0}^{1} x^{2} (x + 1)^{6} d x

\frac{2815}{252}

十進法表記

11.17063 \dots

解答ステップ

\int_{0}^{1} x^{2} (x + 1)^{6} d x

拡張

x^{2} (x + 1)^{6} : x^{8} + 6 x^{7} + 15 x^{6} + 20 x^{5} + 15 x^{4} + 6 x^{3} + x^{2}

= \int_{0}^{1} x^{8} + 6 x^{7} + 15 x^{6} + 20 x^{5} + 15 x^{4} + 6 x^{3} + x^{2} 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} x^{8} d x + \int_{0}^{1} 6 x^{7} d x + \int_{0}^{1} 15 x^{6} d x + \int_{0}^{1} 20 x^{5} d x + \int_{0}^{1} 15 x^{4} d x + \int_{0}^{1} 6 x^{3} d x + \int_{0}^{1} x^{2} d x

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

\int_{0}^{1} 6 x^{7} d x = \frac{3}{4}

\int_{0}^{1} 15 x^{6} d x = \frac{15}{7}

\int_{0}^{1} 20 x^{5} d x = \frac{10}{3}

\int_{0}^{1} 15 x^{4} d x = 3

\int_{0}^{1} 6 x^{3} d x = \frac{3}{2}

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

= \frac{1}{9} + \frac{3}{4} + \frac{15}{7} + \frac{10}{3} + 3 + \frac{3}{2} + \frac{1}{3}

簡素化

\frac{1}{9} + \frac{3}{4} + \frac{15}{7} + \frac{10}{3} + 3 + \frac{3}{2} + \frac{1}{3} : \frac{2815}{252}

= \frac{2815}{252}