integral of x^3(3x^4+2)^6

解答

\int x^{3} (3 x^{4} + 2)^{6} d x

\frac{729 x ^{28}}{28} + \frac{243 x ^{24}}{2} + 243 x^{20} + 270 x^{16} + 180 x^{12} + 72 x^{8} + 16 x^{4} + C

求解步骤

\int x^{3} (3 x^{4} + 2)^{6} d x

乘开

x^{3} (3 x^{4} + 2)^{6} : 729 x^{27} + 2916 x^{23} + 4860 x^{19} + 4320 x^{15} + 2160 x^{11} + 576 x^{7} + 64 x^{3}

= \int 729 x^{27} + 2916 x^{23} + 4860 x^{19} + 4320 x^{15} + 2160 x^{11} + 576 x^{7} + 64 x^{3} 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 729 x^{27} d x + \int 2916 x^{23} d x + \int 4860 x^{19} d x + \int 4320 x^{15} d x + \int 2160 x^{11} d x + \int 576 x^{7} d x + \int 64 x^{3} d x

\int 729 x^{27} d x = \frac{729 x ^{28}}{28}

\int 2916 x^{23} d x = \frac{243 x ^{24}}{2}

\int 4860 x^{19} d x = 243 x^{20}

\int 4320 x^{15} d x = 270 x^{16}

\int 2160 x^{11} d x = 180 x^{12}

\int 576 x^{7} d x = 72 x^{8}

\int 64 x^{3} d x = 16 x^{4}

= \frac{729 x ^{28}}{28} + \frac{243 x ^{24}}{2} + 243 x^{20} + 270 x^{16} + 180 x^{12} + 72 x^{8} + 16 x^{4}

解答补常数

= \frac{729 x ^{28}}{28} + \frac{243 x ^{24}}{2} + 243 x^{20} + 270 x^{16} + 180 x^{12} + 72 x^{8} + 16 x^{4} + C

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