integral of x^3(x^2+1)^{12}

解答

\int x^{3} (x^{2} + 1)^{12} d x

\frac{x ^{28}}{28} + \frac{6 x ^{26}}{13} + \frac{11 x ^{24}}{4} + 10 x^{22} + \frac{99 x ^{20}}{4} + 44 x^{18} + \frac{231 x ^{16}}{4} + \frac{396 x ^{14}}{7} + \frac{165 x ^{12}}{4} + 22 x^{10} + \frac{33 x ^{8}}{4} + 2 x^{6} + \frac{x ^{4}}{4} + C

求解步骤

\int x^{3} (x^{2} + 1)^{12} d x

乘开

x^{3} (x^{2} + 1)^{12} : x^{27} + 12 x^{25} + 66 x^{23} + 220 x^{21} + 495 x^{19} + 792 x^{17} + 924 x^{15} + 792 x^{13} + 495 x^{11} + 220 x^{9} + 66 x^{7} + 12 x^{5} + x^{3}

= \int x^{27} + 12 x^{25} + 66 x^{23} + 220 x^{21} + 495 x^{19} + 792 x^{17} + 924 x^{15} + 792 x^{13} + 495 x^{11} + 220 x^{9} + 66 x^{7} + 12 x^{5} + 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 x^{27} d x + \int 12 x^{25} d x + \int 66 x^{23} d x + \int 220 x^{21} d x + \int 495 x^{19} d x + \int 792 x^{17} d x + \int 924 x^{15} d x + \int 792 x^{13} d x + \int 495 x^{11} d x + \int 220 x^{9} d x + \int 66 x^{7} d x + \int 12 x^{5} d x + \int x^{3} d x

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

\int 12 x^{25} d x = \frac{6 x ^{26}}{13}

\int 66 x^{23} d x = \frac{11 x ^{24}}{4}

\int 220 x^{21} d x = 10 x^{22}

\int 495 x^{19} d x = \frac{99 x ^{20}}{4}

\int 792 x^{17} d x = 44 x^{18}

\int 924 x^{15} d x = \frac{231 x ^{16}}{4}

\int 792 x^{13} d x = \frac{396 x ^{14}}{7}

\int 495 x^{11} d x = \frac{165 x ^{12}}{4}

\int 220 x^{9} d x = 22 x^{10}

\int 66 x^{7} d x = \frac{33 x ^{8}}{4}

\int 12 x^{5} d x = 2 x^{6}

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

= \frac{x ^{28}}{28} + \frac{6 x ^{26}}{13} + \frac{11 x ^{24}}{4} + 10 x^{22} + \frac{99 x ^{20}}{4} + 44 x^{18} + \frac{231 x ^{16}}{4} + \frac{396 x ^{14}}{7} + \frac{165 x ^{12}}{4} + 22 x^{10} + \frac{33 x ^{8}}{4} + 2 x^{6} + \frac{x ^{4}}{4}

解答补常数

= \frac{x ^{28}}{28} + \frac{6 x ^{26}}{13} + \frac{11 x ^{24}}{4} + 10 x^{22} + \frac{99 x ^{20}}{4} + 44 x^{18} + \frac{231 x ^{16}}{4} + \frac{396 x ^{14}}{7} + \frac{165 x ^{12}}{4} + 22 x^{10} + \frac{33 x ^{8}}{4} + 2 x^{6} + \frac{x ^{4}}{4} + C

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