integral from 0 to 1 of x^2(5x^3-1)^4

解答

\int_{0}^{1} x^{2} (5 x^{3} - 1)^{4} d x

\frac{41}{3}

十进制

13.66666 \dots

求解步骤

\int_{0}^{1} x^{2} (5 x^{3} - 1)^{4} d x

乘开

x^{2} (5 x^{3} - 1)^{4} : 625 x^{14} - 500 x^{11} + 150 x^{8} - 20 x^{5} + x^{2}

= \int_{0}^{1} 625 x^{14} - 500 x^{11} + 150 x^{8} - 20 x^{5} + 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} 625 x^{14} d x - \int_{0}^{1} 500 x^{11} d x + \int_{0}^{1} 150 x^{8} d x - \int_{0}^{1} 20 x^{5} d x + \int_{0}^{1} x^{2} d x

\int_{0}^{1} 625 x^{14} d x = \frac{125}{3}

\int_{0}^{1} 500 x^{11} d x = \frac{125}{3}

\int_{0}^{1} 150 x^{8} d x = \frac{50}{3}

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

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

= \frac{125}{3} - \frac{125}{3} + \frac{50}{3} - \frac{10}{3} + \frac{1}{3}

化简

\frac{125}{3} - \frac{125}{3} + \frac{50}{3} - \frac{10}{3} + \frac{1}{3} : \frac{41}{3}

= \frac{41}{3}

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