integral from 1 to 2 of x^2(x^3+3)^3

解答

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

\frac{4795}{4}

十进制

1198.75

求解步骤

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

乘开

x^{2} (x^{3} + 3)^{3} : x^{11} + 9 x^{8} + 27 x^{5} + 27 x^{2}

= \int_{1}^{2} x^{11} + 9 x^{8} + 27 x^{5} + 27 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_{1}^{2} x^{11} d x + \int_{1}^{2} 9 x^{8} d x + \int_{1}^{2} 27 x^{5} d x + \int_{1}^{2} 27 x^{2} d x

\int_{1}^{2} x^{11} d x = \frac{1365}{4}

\int_{1}^{2} 9 x^{8} d x = 511

\int_{1}^{2} 27 x^{5} d x = \frac{567}{2}

\int_{1}^{2} 27 x^{2} d x = 63

= \frac{1365}{4} + 511 + \frac{567}{2} + 63

化简

\frac{1365}{4} + 511 + \frac{567}{2} + 63 : \frac{4795}{4}

= \frac{4795}{4}

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