integral of x^3(x^2+1)^{14}

解答

\int x^{3} (x^{2} + 1)^{14} d x

\frac{x ^{32}}{32} + \frac{7 x ^{30}}{15} + \frac{13 x ^{28}}{4} + 14 x^{26} + \frac{1001 x ^{24}}{24} + 91 x^{22} + \frac{3003 x ^{20}}{20} + \frac{572 x ^{18}}{3} + \frac{3003 x ^{16}}{16} + 143 x^{14} + \frac{1001 x ^{12}}{12} + \frac{182 x ^{10}}{5} + \frac{91 x ^{8}}{8} + \frac{7 x ^{6}}{3} + \frac{x ^{4}}{4} + C

求解步骤

\int x^{3} (x^{2} + 1)^{14} d x

乘开

x^{3} (x^{2} + 1)^{14} : x^{31} + 14 x^{29} + 91 x^{27} + 364 x^{25} + 1001 x^{23} + 2002 x^{21} + 3003 x^{19} + 3432 x^{17} + 3003 x^{15} + 2002 x^{13} + 1001 x^{11} + 364 x^{9} + 91 x^{7} + 14 x^{5} + x^{3}

= \int x^{31} + 14 x^{29} + 91 x^{27} + 364 x^{25} + 1001 x^{23} + 2002 x^{21} + 3003 x^{19} + 3432 x^{17} + 3003 x^{15} + 2002 x^{13} + 1001 x^{11} + 364 x^{9} + 91 x^{7} + 14 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^{31} d x + \int 14 x^{29} d x + \int 91 x^{27} d x + \int 364 x^{25} d x + \int 1001 x^{23} d x + \int 2002 x^{21} d x + \int 3003 x^{19} d x + \int 3432 x^{17} d x + \int 3003 x^{15} d x + \int 2002 x^{13} d x + \int 1001 x^{11} d x + \int 364 x^{9} d x + \int 91 x^{7} d x + \int 14 x^{5} d x + \int x^{3} d x

\int x^{31} d x = \frac{x ^{32}}{32}

\int 14 x^{29} d x = \frac{7 x ^{30}}{15}

\int 91 x^{27} d x = \frac{13 x ^{28}}{4}

\int 364 x^{25} d x = 14 x^{26}

\int 1001 x^{23} d x = \frac{1001 x ^{24}}{24}

\int 2002 x^{21} d x = 91 x^{22}

\int 3003 x^{19} d x = \frac{3003 x ^{20}}{20}

\int 3432 x^{17} d x = \frac{572 x ^{18}}{3}

\int 3003 x^{15} d x = \frac{3003 x ^{16}}{16}

\int 2002 x^{13} d x = 143 x^{14}

\int 1001 x^{11} d x = \frac{1001 x ^{12}}{12}

\int 364 x^{9} d x = \frac{182 x ^{10}}{5}

\int 91 x^{7} d x = \frac{91 x ^{8}}{8}

\int 14 x^{5} d x = \frac{7 x ^{6}}{3}

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

= \frac{x ^{32}}{32} + \frac{7 x ^{30}}{15} + \frac{13 x ^{28}}{4} + 14 x^{26} + \frac{1001 x ^{24}}{24} + 91 x^{22} + \frac{3003 x ^{20}}{20} + \frac{572 x ^{18}}{3} + \frac{3003 x ^{16}}{16} + 143 x^{14} + \frac{1001 x ^{12}}{12} + \frac{182 x ^{10}}{5} + \frac{91 x ^{8}}{8} + \frac{7 x ^{6}}{3} + \frac{x ^{4}}{4}

解答补常数

= \frac{x ^{32}}{32} + \frac{7 x ^{30}}{15} + \frac{13 x ^{28}}{4} + 14 x^{26} + \frac{1001 x ^{24}}{24} + 91 x^{22} + \frac{3003 x ^{20}}{20} + \frac{572 x ^{18}}{3} + \frac{3003 x ^{16}}{16} + 143 x^{14} + \frac{1001 x ^{12}}{12} + \frac{182 x ^{10}}{5} + \frac{91 x ^{8}}{8} + \frac{7 x ^{6}}{3} + \frac{x ^{4}}{4} + C

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