integral from 0 to 3 of x(x^2-1)^3

解答

\int_{0}^{3} x (x^{2} - 1)^{3} d x

\frac{4095}{8}

十进制

511.875

求解步骤

\int_{0}^{3} x (x^{2} - 1)^{3} d x

乘开

x (x^{2} - 1)^{3} : x^{7} - 3 x^{5} + 3 x^{3} - x

= \int_{0}^{3} x^{7} - 3 x^{5} + 3 x^{3} - x 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}^{3} x^{7} d x - \int_{0}^{3} 3 x^{5} d x + \int_{0}^{3} 3 x^{3} d x - \int_{0}^{3} x d x

\int_{0}^{3} x^{7} d x = \frac{6561}{8}

\int_{0}^{3} 3 x^{5} d x = \frac{729}{2}

\int_{0}^{3} 3 x^{3} d x = \frac{243}{4}

\int_{0}^{3} x d x = \frac{9}{2}

= \frac{6561}{8} - \frac{729}{2} + \frac{243}{4} - \frac{9}{2}

化简

\frac{6561}{8} - \frac{729}{2} + \frac{243}{4} - \frac{9}{2} : \frac{4095}{8}

= \frac{4095}{8}

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