integral from 0 to 1 of x^2(x+1)^7

解答

\int_{0}^{1} x^{2} (x + 1)^{7} d x

\frac{7423}{360}

十进制

20.61944 \dots

求解步骤

\int_{0}^{1} x^{2} (x + 1)^{7} d x

乘开

x^{2} (x + 1)^{7} : x^{9} + 7 x^{8} + 21 x^{7} + 35 x^{6} + 35 x^{5} + 21 x^{4} + 7 x^{3} + x^{2}

= \int_{0}^{1} x^{9} + 7 x^{8} + 21 x^{7} + 35 x^{6} + 35 x^{5} + 21 x^{4} + 7 x^{3} + x^{2} 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} x^{9} d x + \int_{0}^{1} 7 x^{8} d x + \int_{0}^{1} 21 x^{7} d x + \int_{0}^{1} 35 x^{6} d x + \int_{0}^{1} 35 x^{5} d x + \int_{0}^{1} 21 x^{4} d x + \int_{0}^{1} 7 x^{3} d x + \int_{0}^{1} x^{2} d x

\int_{0}^{1} x^{9} d x = \frac{1}{10}

\int_{0}^{1} 7 x^{8} d x = \frac{7}{9}

\int_{0}^{1} 21 x^{7} d x = \frac{21}{8}

\int_{0}^{1} 35 x^{6} d x = 5

\int_{0}^{1} 35 x^{5} d x = \frac{35}{6}

\int_{0}^{1} 21 x^{4} d x = \frac{21}{5}

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

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

= \frac{1}{10} + \frac{7}{9} + \frac{21}{8} + 5 + \frac{35}{6} + \frac{21}{5} + \frac{7}{4} + \frac{1}{3}

化简

\frac{1}{10} + \frac{7}{9} + \frac{21}{8} + 5 + \frac{35}{6} + \frac{21}{5} + \frac{7}{4} + \frac{1}{3} : \frac{7423}{360}

= \frac{7423}{360}

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