integral of x^3(3-x^4)^{10}

解答

\int x^{3} (3 - x^{4})^{10} d x

\frac{59049 x ^{4}}{4} - \frac{98415 x ^{8}}{4} + \frac{98415 x ^{12}}{4} - \frac{32805 x ^{16}}{2} + \frac{15309 x ^{20}}{2} - \frac{5103 x ^{24}}{2} + \frac{1215 x ^{28}}{2} - \frac{405 x ^{32}}{4} + \frac{45 x ^{36}}{4} - \frac{3 x ^{40}}{4} + \frac{x ^{44}}{44} + C

求解步骤

\int x^{3} (3 - x^{4})^{10} d x

乘开

x^{3} (3 - x^{4})^{10} : 59049 x^{3} - 196830 x^{7} + 295245 x^{11} - 262440 x^{15} + 153090 x^{19} - 61236 x^{23} + 17010 x^{27} - 3240 x^{31} + 405 x^{35} - 30 x^{39} + x^{43}

= \int 59049 x^{3} - 196830 x^{7} + 295245 x^{11} - 262440 x^{15} + 153090 x^{19} - 61236 x^{23} + 17010 x^{27} - 3240 x^{31} + 405 x^{35} - 30 x^{39} + x^{43} 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 59049 x^{3} d x - \int 196830 x^{7} d x + \int 295245 x^{11} d x - \int 262440 x^{15} d x + \int 153090 x^{19} d x - \int 61236 x^{23} d x + \int 17010 x^{27} d x - \int 3240 x^{31} d x + \int 405 x^{35} d x - \int 30 x^{39} d x + \int x^{43} d x

\int 59049 x^{3} d x = \frac{59049 x ^{4}}{4}

\int 196830 x^{7} d x = \frac{98415 x ^{8}}{4}

\int 295245 x^{11} d x = \frac{98415 x ^{12}}{4}

\int 262440 x^{15} d x = \frac{32805 x ^{16}}{2}

\int 153090 x^{19} d x = \frac{15309 x ^{20}}{2}

\int 61236 x^{23} d x = \frac{5103 x ^{24}}{2}

\int 17010 x^{27} d x = \frac{1215 x ^{28}}{2}

\int 3240 x^{31} d x = \frac{405 x ^{32}}{4}

\int 405 x^{35} d x = \frac{45 x ^{36}}{4}

\int 30 x^{39} d x = \frac{3 x ^{40}}{4}

\int x^{43} d x = \frac{x ^{44}}{44}

= \frac{59049 x ^{4}}{4} - \frac{98415 x ^{8}}{4} + \frac{98415 x ^{12}}{4} - \frac{32805 x ^{16}}{2} + \frac{15309 x ^{20}}{2} - \frac{5103 x ^{24}}{2} + \frac{1215 x ^{28}}{2} - \frac{405 x ^{32}}{4} + \frac{45 x ^{36}}{4} - \frac{3 x ^{40}}{4} + \frac{x ^{44}}{44}

解答补常数

= \frac{59049 x ^{4}}{4} - \frac{98415 x ^{8}}{4} + \frac{98415 x ^{12}}{4} - \frac{32805 x ^{16}}{2} + \frac{15309 x ^{20}}{2} - \frac{5103 x ^{24}}{2} + \frac{1215 x ^{28}}{2} - \frac{405 x ^{32}}{4} + \frac{45 x ^{36}}{4} - \frac{3 x ^{40}}{4} + \frac{x ^{44}}{44} + C

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