integral of (4x^3-3x^2)(4x^4-4x^3)^5

解答

\int (4 x^{3} - 3 x^{2}) (4 x^{4} - 4 x^{3})^{5} d x

\frac{512 x ^{24}}{3} - 1024 x^{23} + 2560 x^{22} - \frac{10240 x ^{21}}{3} + 2560 x^{20} - 1024 x^{19} + \frac{512 x ^{18}}{3} + C

求解步骤

\int (4 x^{3} - 3 x^{2}) (4 x^{4} - 4 x^{3})^{5} d x

乘开

(4 x^{3} - 3 x^{2}) (4 x^{4} - 4 x^{3})^{5} : 4096 x^{23} - 23552 x^{22} + 56320 x^{21} - 71680 x^{20} + 51200 x^{19} - 19456 x^{18} + 3072 x^{17}

= \int 4096 x^{23} - 23552 x^{22} + 56320 x^{21} - 71680 x^{20} + 51200 x^{19} - 19456 x^{18} + 3072 x^{17} 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 4096 x^{23} d x - \int 23552 x^{22} d x + \int 56320 x^{21} d x - \int 71680 x^{20} d x + \int 51200 x^{19} d x - \int 19456 x^{18} d x + \int 3072 x^{17} d x

\int 4096 x^{23} d x = \frac{512 x ^{24}}{3}

\int 23552 x^{22} d x = 1024 x^{23}

\int 56320 x^{21} d x = 2560 x^{22}

\int 71680 x^{20} d x = \frac{10240 x ^{21}}{3}

\int 51200 x^{19} d x = 2560 x^{20}

\int 19456 x^{18} d x = 1024 x^{19}

\int 3072 x^{17} d x = \frac{512 x ^{18}}{3}

= \frac{512 x ^{24}}{3} - 1024 x^{23} + 2560 x^{22} - \frac{10240 x ^{21}}{3} + 2560 x^{20} - 1024 x^{19} + \frac{512 x ^{18}}{3}

解答补常数

= \frac{512 x ^{24}}{3} - 1024 x^{23} + 2560 x^{22} - \frac{10240 x ^{21}}{3} + 2560 x^{20} - 1024 x^{19} + \frac{512 x ^{18}}{3} + C

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