integral of x^2(4x^3-3)^8

解答

\int x^{2} (4 x^{3} - 3)^{8} d x

\frac{65536 x ^{27}}{27} - 16384 x^{24} + 49152 x^{21} - 86016 x^{18} + 96768 x^{15} - 72576 x^{12} + 36288 x^{9} - 11664 x^{6} + 2187 x^{3} + C

求解步骤

\int x^{2} (4 x^{3} - 3)^{8} d x

因式分解

4 x^{3} - 3 : - (- 4 x^{3} + 3)

= \int x^{2} (- (- 4 x^{3} + 3))^{8} d x

(- (- 4 x^{3} + 3))^{8} = (- 4 x^{3} + 3)^{8}

= \int x^{2} (- 4 x^{3} + 3)^{8} d x

乘开

x^{2} (- 4 x^{3} + 3)^{8} : 65536 x^{26} - 393216 x^{23} + 1032192 x^{20} - 1548288 x^{17} + 1451520 x^{14} - 870912 x^{11} + 326592 x^{8} - 69984 x^{5} + 6561 x^{2}

= \int 65536 x^{26} - 393216 x^{23} + 1032192 x^{20} - 1548288 x^{17} + 1451520 x^{14} - 870912 x^{11} + 326592 x^{8} - 69984 x^{5} + 6561 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 65536 x^{26} d x - \int 393216 x^{23} d x + \int 1032192 x^{20} d x - \int 1548288 x^{17} d x + \int 1451520 x^{14} d x - \int 870912 x^{11} d x + \int 326592 x^{8} d x - \int 69984 x^{5} d x + \int 6561 x^{2} d x

\int 65536 x^{26} d x = \frac{65536 x ^{27}}{27}

\int 393216 x^{23} d x = 16384 x^{24}

\int 1032192 x^{20} d x = 49152 x^{21}

\int 1548288 x^{17} d x = 86016 x^{18}

\int 1451520 x^{14} d x = 96768 x^{15}

\int 870912 x^{11} d x = 72576 x^{12}

\int 326592 x^{8} d x = 36288 x^{9}

\int 69984 x^{5} d x = 11664 x^{6}

\int 6561 x^{2} d x = 2187 x^{3}

= \frac{65536 x ^{27}}{27} - 16384 x^{24} + 49152 x^{21} - 86016 x^{18} + 96768 x^{15} - 72576 x^{12} + 36288 x^{9} - 11664 x^{6} + 2187 x^{3}

解答补常数

= \frac{65536 x ^{27}}{27} - 16384 x^{24} + 49152 x^{21} - 86016 x^{18} + 96768 x^{15} - 72576 x^{12} + 36288 x^{9} - 11664 x^{6} + 2187 x^{3} + C

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