integral of x^2(5x^3-3)^7

解答

\int x^{2} (5 x^{3} - 3)^{7} d x

\frac{78125 x ^{24}}{24} - 15625 x^{21} + \frac{65625 x ^{18}}{2} - 39375 x^{15} + \frac{118125 x ^{12}}{4} - 14175 x^{9} + \frac{8505 x ^{6}}{2} - 729 x^{3} + C

求解步骤

\int x^{2} (5 x^{3} - 3)^{7} d x

乘开

x^{2} (5 x^{3} - 3)^{7} : 78125 x^{23} - 328125 x^{20} + 590625 x^{17} - 590625 x^{14} + 354375 x^{11} - 127575 x^{8} + 25515 x^{5} - 2187 x^{2}

= \int 78125 x^{23} - 328125 x^{20} + 590625 x^{17} - 590625 x^{14} + 354375 x^{11} - 127575 x^{8} + 25515 x^{5} - 2187 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 78125 x^{23} d x - \int 328125 x^{20} d x + \int 590625 x^{17} d x - \int 590625 x^{14} d x + \int 354375 x^{11} d x - \int 127575 x^{8} d x + \int 25515 x^{5} d x - \int 2187 x^{2} d x

\int 78125 x^{23} d x = \frac{78125 x ^{24}}{24}

\int 328125 x^{20} d x = 15625 x^{21}

\int 590625 x^{17} d x = \frac{65625 x ^{18}}{2}

\int 590625 x^{14} d x = 39375 x^{15}

\int 354375 x^{11} d x = \frac{118125 x ^{12}}{4}

\int 127575 x^{8} d x = 14175 x^{9}

\int 25515 x^{5} d x = \frac{8505 x ^{6}}{2}

\int 2187 x^{2} d x = 729 x^{3}

= \frac{78125 x ^{24}}{24} - 15625 x^{21} + \frac{65625 x ^{18}}{2} - 39375 x^{15} + \frac{118125 x ^{12}}{4} - 14175 x^{9} + \frac{8505 x ^{6}}{2} - 729 x^{3}

解答补常数

= \frac{78125 x ^{24}}{24} - 15625 x^{21} + \frac{65625 x ^{18}}{2} - 39375 x^{15} + \frac{118125 x ^{12}}{4} - 14175 x^{9} + \frac{8505 x ^{6}}{2} - 729 x^{3} + C

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