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

解答

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

\frac{x ^{14}}{14} + \frac{24 x ^{13}}{13} + 22 x^{12} + 160 x^{11} + 792 x^{10} + 2816 x^{9} + 7392 x^{8} + \frac{101376 x ^{7}}{7} + 21120 x^{6} + 22528 x^{5} + 16896 x^{4} + 8192 x^{3} + 2048 x^{2} + C

求解步骤

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

乘开

x (x + 2)^{12} : x^{13} + 24 x^{12} + 264 x^{11} + 1760 x^{10} + 7920 x^{9} + 25344 x^{8} + 59136 x^{7} + 101376 x^{6} + 126720 x^{5} + 112640 x^{4} + 67584 x^{3} + 24576 x^{2} + 4096 x

= \int x^{13} + 24 x^{12} + 264 x^{11} + 1760 x^{10} + 7920 x^{9} + 25344 x^{8} + 59136 x^{7} + 101376 x^{6} + 126720 x^{5} + 112640 x^{4} + 67584 x^{3} + 24576 x^{2} + 4096 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 24 x^{12} d x + \int 264 x^{11} d x + \int 1760 x^{10} d x + \int 7920 x^{9} d x + \int 25344 x^{8} d x + \int 59136 x^{7} d x + \int 101376 x^{6} d x + \int 126720 x^{5} d x + \int 112640 x^{4} d x + \int 67584 x^{3} d x + \int 24576 x^{2} d x + \int 4096 x d x

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

\int 24 x^{12} d x = \frac{24 x ^{13}}{13}

\int 264 x^{11} d x = 22 x^{12}

\int 1760 x^{10} d x = 160 x^{11}

\int 7920 x^{9} d x = 792 x^{10}

\int 25344 x^{8} d x = 2816 x^{9}

\int 59136 x^{7} d x = 7392 x^{8}

\int 101376 x^{6} d x = \frac{101376 x ^{7}}{7}

\int 126720 x^{5} d x = 21120 x^{6}

\int 112640 x^{4} d x = 22528 x^{5}

\int 67584 x^{3} d x = 16896 x^{4}

\int 24576 x^{2} d x = 8192 x^{3}

\int 4096 x d x = 2048 x^{2}

= \frac{x ^{14}}{14} + \frac{24 x ^{13}}{13} + 22 x^{12} + 160 x^{11} + 792 x^{10} + 2816 x^{9} + 7392 x^{8} + \frac{101376 x ^{7}}{7} + 21120 x^{6} + 22528 x^{5} + 16896 x^{4} + 8192 x^{3} + 2048 x^{2}

解答补常数

= \frac{x ^{14}}{14} + \frac{24 x ^{13}}{13} + 22 x^{12} + 160 x^{11} + 792 x^{10} + 2816 x^{9} + 7392 x^{8} + \frac{101376 x ^{7}}{7} + 21120 x^{6} + 22528 x^{5} + 16896 x^{4} + 8192 x^{3} + 2048 x^{2} + C