integral of x(x-5)^{12}

解答

\int x (x - 5)^{12} d x

\frac{x ^{14}}{14} - \frac{60 x ^{13}}{13} + \frac{275 x ^{12}}{2} - 2500 x^{11} + \frac{61875 x ^{10}}{2} - 275000 x^{9} + \frac{3609375 x ^{8}}{2} - \frac{61875000 x ^{7}}{7} + \frac{64453125 x ^{6}}{2} - 85937500 x^{5} + \frac{322265625 x ^{4}}{2} - 195312500 x^{3} + \frac{244140625 x ^{2}}{2} + C

求解步骤

\int x (x - 5)^{12} d x

乘开

x (x - 5)^{12} : x^{13} - 60 x^{12} + 1650 x^{11} - 27500 x^{10} + 309375 x^{9} - 2475000 x^{8} + 14437500 x^{7} - 61875000 x^{6} + 193359375 x^{5} - 429687500 x^{4} + 644531250 x^{3} - 585937500 x^{2} + 244140625 x

= \int x^{13} - 60 x^{12} + 1650 x^{11} - 27500 x^{10} + 309375 x^{9} - 2475000 x^{8} + 14437500 x^{7} - 61875000 x^{6} + 193359375 x^{5} - 429687500 x^{4} + 644531250 x^{3} - 585937500 x^{2} + 244140625 x 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^{13} d x - \int 60 x^{12} d x + \int 1650 x^{11} d x - \int 27500 x^{10} d x + \int 309375 x^{9} d x - \int 2475000 x^{8} d x + \int 14437500 x^{7} d x - \int 61875000 x^{6} d x + \int 193359375 x^{5} d x - \int 429687500 x^{4} d x + \int 644531250 x^{3} d x - \int 585937500 x^{2} d x + \int 244140625 x d x

\int x^{13} d x = \frac{x ^{14}}{14}

\int 60 x^{12} d x = \frac{60 x ^{13}}{13}

\int 1650 x^{11} d x = \frac{275 x ^{12}}{2}

\int 27500 x^{10} d x = 2500 x^{11}

\int 309375 x^{9} d x = \frac{61875 x ^{10}}{2}

\int 2475000 x^{8} d x = 275000 x^{9}

\int 14437500 x^{7} d x = \frac{3609375 x ^{8}}{2}

\int 61875000 x^{6} d x = \frac{61875000 x ^{7}}{7}

\int 193359375 x^{5} d x = \frac{64453125 x ^{6}}{2}

\int 429687500 x^{4} d x = 85937500 x^{5}

\int 644531250 x^{3} d x = \frac{322265625 x ^{4}}{2}

\int 585937500 x^{2} d x = 195312500 x^{3}

\int 244140625 x d x = \frac{244140625 x ^{2}}{2}

= \frac{x ^{14}}{14} - \frac{60 x ^{13}}{13} + \frac{275 x ^{12}}{2} - 2500 x^{11} + \frac{61875 x ^{10}}{2} - 275000 x^{9} + \frac{3609375 x ^{8}}{2} - \frac{61875000 x ^{7}}{7} + \frac{64453125 x ^{6}}{2} - 85937500 x^{5} + \frac{322265625 x ^{4}}{2} - 195312500 x^{3} + \frac{244140625 x ^{2}}{2}

解答补常数

= \frac{x ^{14}}{14} - \frac{60 x ^{13}}{13} + \frac{275 x ^{12}}{2} - 2500 x^{11} + \frac{61875 x ^{10}}{2} - 275000 x^{9} + \frac{3609375 x ^{8}}{2} - \frac{61875000 x ^{7}}{7} + \frac{64453125 x ^{6}}{2} - 85937500 x^{5} + \frac{322265625 x ^{4}}{2} - 195312500 x^{3} + \frac{244140625 x ^{2}}{2} + C

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