integral of (x^3+3x)^8(x^2+1)

解答

\int (x^{3} + 3 x)^{8} (x^{2} + 1) d x

\frac{x ^{27}}{27} + x^{25} + 12 x^{23} + 84 x^{21} + 378 x^{19} + 1134 x^{17} + 2268 x^{15} + 2916 x^{13} + 2187 x^{11} + 729 x^{9} + C

求解步骤

\int (x^{3} + 3 x)^{8} (x^{2} + 1) d x

乘开

(x^{3} + 3 x)^{8} (x^{2} + 1) : x^{26} + 25 x^{24} + 276 x^{22} + 1764 x^{20} + 7182 x^{18} + 19278 x^{16} + 34020 x^{14} + 37908 x^{12} + 24057 x^{10} + 6561 x^{8}

= \int x^{26} + 25 x^{24} + 276 x^{22} + 1764 x^{20} + 7182 x^{18} + 19278 x^{16} + 34020 x^{14} + 37908 x^{12} + 24057 x^{10} + 6561 x^{8} 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 x^{26} d x + \int 25 x^{24} d x + \int 276 x^{22} d x + \int 1764 x^{20} d x + \int 7182 x^{18} d x + \int 19278 x^{16} d x + \int 34020 x^{14} d x + \int 37908 x^{12} d x + \int 24057 x^{10} d x + \int 6561 x^{8} d x

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

\int 25 x^{24} d x = x^{25}

\int 276 x^{22} d x = 12 x^{23}

\int 1764 x^{20} d x = 84 x^{21}

\int 7182 x^{18} d x = 378 x^{19}

\int 19278 x^{16} d x = 1134 x^{17}

\int 34020 x^{14} d x = 2268 x^{15}

\int 37908 x^{12} d x = 2916 x^{13}

\int 24057 x^{10} d x = 2187 x^{11}

\int 6561 x^{8} d x = 729 x^{9}

= \frac{x ^{27}}{27} + x^{25} + 12 x^{23} + 84 x^{21} + 378 x^{19} + 1134 x^{17} + 2268 x^{15} + 2916 x^{13} + 2187 x^{11} + 729 x^{9}

解答补常数

= \frac{x ^{27}}{27} + x^{25} + 12 x^{23} + 84 x^{21} + 378 x^{19} + 1134 x^{17} + 2268 x^{15} + 2916 x^{13} + 2187 x^{11} + 729 x^{9} + C

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