integral of x(1-x)^{15}

解答

\int x (1 - x)^{15} d x

\frac{x ^{2}}{2} - 5 x^{3} + \frac{105 x ^{4}}{4} - 91 x^{5} + \frac{455 x ^{6}}{2} - 429 x^{7} + \frac{5005 x ^{8}}{8} - 715 x^{9} + \frac{1287 x ^{10}}{2} - 455 x^{11} + \frac{1001 x ^{12}}{4} - 105 x^{13} + \frac{65 x ^{14}}{2} - 7 x^{15} + \frac{15 x ^{16}}{16} - \frac{x ^{17}}{17} + C

求解步骤

\int x (1 - x)^{15} d x

乘开

x (1 - x)^{15} : x - 15 x^{2} + 105 x^{3} - 455 x^{4} + 1365 x^{5} - 3003 x^{6} + 5005 x^{7} - 6435 x^{8} + 6435 x^{9} - 5005 x^{10} + 3003 x^{11} - 1365 x^{12} + 455 x^{13} - 105 x^{14} + 15 x^{15} - x^{16}

= \int x - 15 x^{2} + 105 x^{3} - 455 x^{4} + 1365 x^{5} - 3003 x^{6} + 5005 x^{7} - 6435 x^{8} + 6435 x^{9} - 5005 x^{10} + 3003 x^{11} - 1365 x^{12} + 455 x^{13} - 105 x^{14} + 15 x^{15} - x^{16} 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 d x - \int 15 x^{2} d x + \int 105 x^{3} d x - \int 455 x^{4} d x + \int 1365 x^{5} d x - \int 3003 x^{6} d x + \int 5005 x^{7} d x - \int 6435 x^{8} d x + \int 6435 x^{9} d x - \int 5005 x^{10} d x + \int 3003 x^{11} d x - \int 1365 x^{12} d x + \int 455 x^{13} d x - \int 105 x^{14} d x + \int 15 x^{15} d x - \int x^{16} d x

\int x d x = \frac{x ^{2}}{2}

\int 15 x^{2} d x = 5 x^{3}

\int 105 x^{3} d x = \frac{105 x ^{4}}{4}

\int 455 x^{4} d x = 91 x^{5}

\int 1365 x^{5} d x = \frac{455 x ^{6}}{2}

\int 3003 x^{6} d x = 429 x^{7}

\int 5005 x^{7} d x = \frac{5005 x ^{8}}{8}

\int 6435 x^{8} d x = 715 x^{9}

\int 6435 x^{9} d x = \frac{1287 x ^{10}}{2}

\int 5005 x^{10} d x = 455 x^{11}

\int 3003 x^{11} d x = \frac{1001 x ^{12}}{4}

\int 1365 x^{12} d x = 105 x^{13}

\int 455 x^{13} d x = \frac{65 x ^{14}}{2}

\int 105 x^{14} d x = 7 x^{15}

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

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

= \frac{x ^{2}}{2} - 5 x^{3} + \frac{105 x ^{4}}{4} - 91 x^{5} + \frac{455 x ^{6}}{2} - 429 x^{7} + \frac{5005 x ^{8}}{8} - 715 x^{9} + \frac{1287 x ^{10}}{2} - 455 x^{11} + \frac{1001 x ^{12}}{4} - 105 x^{13} + \frac{65 x ^{14}}{2} - 7 x^{15} + \frac{15 x ^{16}}{16} - \frac{x ^{17}}{17}

解答补常数

= \frac{x ^{2}}{2} - 5 x^{3} + \frac{105 x ^{4}}{4} - 91 x^{5} + \frac{455 x ^{6}}{2} - 429 x^{7} + \frac{5005 x ^{8}}{8} - 715 x^{9} + \frac{1287 x ^{10}}{2} - 455 x^{11} + \frac{1001 x ^{12}}{4} - 105 x^{13} + \frac{65 x ^{14}}{2} - 7 x^{15} + \frac{15 x ^{16}}{16} - \frac{x ^{17}}{17} + C

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