integral of (6x^3+5x)^{12}

解答

\int (6 x^{3} + 5 x)^{12} d x

\frac{6 ^{12} x ^{37}}{37} + \frac{4353564672 x ^{35}}{7} + 3023308800 x^{33} + \frac{277136640000 x ^{31}}{31} + \frac{519631200000 x ^{29}}{29} + 25660800000 x^{27} + 26943840000 x^{25} + \frac{481140000000 x ^{23}}{23} + \frac{83531250000 x ^{21}}{7} + \frac{92812500000 x ^{19}}{19} + \frac{23203125000 x ^{17}}{17} + 234375000 x^{15} + \frac{244140625 x ^{13}}{13} + C

求解步骤

\int (6 x^{3} + 5 x)^{12} d x

乘开

(6 x^{3} + 5 x)^{12} : 6^{12} x^{36} + 21767823360 x^{34} + 99769190400 x^{32} + 277136640000 x^{30} + 519631200000 x^{28} + 692841600000 x^{26} + 673596000000 x^{24} + 481140000000 x^{22} + 250593750000 x^{20} + 92812500000 x^{18} + 23203125000 x^{16} + 3515625000 x^{14} + 244140625 x^{12}

= \int 6^{12} x^{36} + 21767823360 x^{34} + 99769190400 x^{32} + 277136640000 x^{30} + 519631200000 x^{28} + 692841600000 x^{26} + 673596000000 x^{24} + 481140000000 x^{22} + 250593750000 x^{20} + 92812500000 x^{18} + 23203125000 x^{16} + 3515625000 x^{14} + 244140625 x^{12} 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 6^{12} x^{36} d x + \int 21767823360 x^{34} d x + \int 99769190400 x^{32} d x + \int 277136640000 x^{30} d x + \int 519631200000 x^{28} d x + \int 692841600000 x^{26} d x + \int 673596000000 x^{24} d x + \int 481140000000 x^{22} d x + \int 250593750000 x^{20} d x + \int 92812500000 x^{18} d x + \int 23203125000 x^{16} d x + \int 3515625000 x^{14} d x + \int 244140625 x^{12} d x

\int 6^{12} x^{36} d x = \frac{6 ^{12} x ^{37}}{37}

\int 21767823360 x^{34} d x = \frac{4353564672 x ^{35}}{7}

\int 99769190400 x^{32} d x = 3023308800 x^{33}

\int 277136640000 x^{30} d x = \frac{277136640000 x ^{31}}{31}

\int 519631200000 x^{28} d x = \frac{519631200000 x ^{29}}{29}

\int 692841600000 x^{26} d x = 25660800000 x^{27}

\int 673596000000 x^{24} d x = 26943840000 x^{25}

\int 481140000000 x^{22} d x = \frac{481140000000 x ^{23}}{23}

\int 250593750000 x^{20} d x = \frac{83531250000 x ^{21}}{7}

\int 92812500000 x^{18} d x = \frac{92812500000 x ^{19}}{19}

\int 23203125000 x^{16} d x = \frac{23203125000 x ^{17}}{17}

\int 3515625000 x^{14} d x = 234375000 x^{15}

\int 244140625 x^{12} d x = \frac{244140625 x ^{13}}{13}

= \frac{6 ^{12} x ^{37}}{37} + \frac{4353564672 x ^{35}}{7} + 3023308800 x^{33} + \frac{277136640000 x ^{31}}{31} + \frac{519631200000 x ^{29}}{29} + 25660800000 x^{27} + 26943840000 x^{25} + \frac{481140000000 x ^{23}}{23} + \frac{83531250000 x ^{21}}{7} + \frac{92812500000 x ^{19}}{19} + \frac{23203125000 x ^{17}}{17} + 234375000 x^{15} + \frac{244140625 x ^{13}}{13}

解答补常数

= \frac{6 ^{12} x ^{37}}{37} + \frac{4353564672 x ^{35}}{7} + 3023308800 x^{33} + \frac{277136640000 x ^{31}}{31} + \frac{519631200000 x ^{29}}{29} + 25660800000 x^{27} + 26943840000 x^{25} + \frac{481140000000 x ^{23}}{23} + \frac{83531250000 x ^{21}}{7} + \frac{92812500000 x ^{19}}{19} + \frac{23203125000 x ^{17}}{17} + 234375000 x^{15} + \frac{244140625 x ^{13}}{13} + C

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