integral of (2x^2+2x-x)^{10}(2x+1)

解答

\int (2 x^{2} + 2 x - x)^{10} (2 x + 1) d x

\frac{1024 x ^{22}}{11} + \frac{11264 x ^{21}}{21} + 1408 x^{20} + \frac{42240 x ^{19}}{19} + \frac{7040 x ^{18}}{3} + \frac{29568 x ^{17}}{17} + 924 x^{16} + 352 x^{15} + \frac{660 x ^{14}}{7} + \frac{220 x ^{13}}{13} + \frac{11 x ^{12}}{6} + \frac{x ^{11}}{11} + C

求解步骤

\int (2 x^{2} + 2 x - x)^{10} (2 x + 1) d x

乘开

(2 x^{2} + 2 x - x)^{10} (2 x + 1) : 2048 x^{21} + 11264 x^{20} + 28160 x^{19} + 42240 x^{18} + 42240 x^{17} + 29568 x^{16} + 14784 x^{15} + 5280 x^{14} + 1320 x^{13} + 220 x^{12} + 22 x^{11} + x^{10}

= \int 2048 x^{21} + 11264 x^{20} + 28160 x^{19} + 42240 x^{18} + 42240 x^{17} + 29568 x^{16} + 14784 x^{15} + 5280 x^{14} + 1320 x^{13} + 220 x^{12} + 22 x^{11} + x^{10} 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 2048 x^{21} d x + \int 11264 x^{20} d x + \int 28160 x^{19} d x + \int 42240 x^{18} d x + \int 42240 x^{17} d x + \int 29568 x^{16} d x + \int 14784 x^{15} d x + \int 5280 x^{14} d x + \int 1320 x^{13} d x + \int 220 x^{12} d x + \int 22 x^{11} d x + \int x^{10} d x

\int 2048 x^{21} d x = \frac{1024 x ^{22}}{11}

\int 11264 x^{20} d x = \frac{11264 x ^{21}}{21}

\int 28160 x^{19} d x = 1408 x^{20}

\int 42240 x^{18} d x = \frac{42240 x ^{19}}{19}

\int 42240 x^{17} d x = \frac{7040 x ^{18}}{3}

\int 29568 x^{16} d x = \frac{29568 x ^{17}}{17}

\int 14784 x^{15} d x = 924 x^{16}

\int 5280 x^{14} d x = 352 x^{15}

\int 1320 x^{13} d x = \frac{660 x ^{14}}{7}

\int 220 x^{12} d x = \frac{220 x ^{13}}{13}

\int 22 x^{11} d x = \frac{11 x ^{12}}{6}

\int x^{10} d x = \frac{x ^{11}}{11}

= \frac{1024 x ^{22}}{11} + \frac{11264 x ^{21}}{21} + 1408 x^{20} + \frac{42240 x ^{19}}{19} + \frac{7040 x ^{18}}{3} + \frac{29568 x ^{17}}{17} + 924 x^{16} + 352 x^{15} + \frac{660 x ^{14}}{7} + \frac{220 x ^{13}}{13} + \frac{11 x ^{12}}{6} + \frac{x ^{11}}{11}

解答补常数

= \frac{1024 x ^{22}}{11} + \frac{11264 x ^{21}}{21} + 1408 x^{20} + \frac{42240 x ^{19}}{19} + \frac{7040 x ^{18}}{3} + \frac{29568 x ^{17}}{17} + 924 x^{16} + 352 x^{15} + \frac{660 x ^{14}}{7} + \frac{220 x ^{13}}{13} + \frac{11 x ^{12}}{6} + \frac{x ^{11}}{11} + C

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