integral of (3-3x)^8

解答

\int (3 - 3 x)^{8} d x

6561 x - 26244 x^{2} + 61236 x^{3} - 91854 x^{4} + 91854 x^{5} - 61236 x^{6} + 26244 x^{7} - 6561 x^{8} + 729 x^{9} + C

求解步骤

\int (3 - 3 x)^{8} d x

乘开

(3 - 3 x)^{8} : 6561 - 52488 x + 183708 x^{2} - 367416 x^{3} + 459270 x^{4} - 367416 x^{5} + 183708 x^{6} - 52488 x^{7} + 6561 x^{8}

= \int 6561 - 52488 x + 183708 x^{2} - 367416 x^{3} + 459270 x^{4} - 367416 x^{5} + 183708 x^{6} - 52488 x^{7} + 6561 x^{8} 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 6561 d x - \int 52488 x d x + \int 183708 x^{2} d x - \int 367416 x^{3} d x + \int 459270 x^{4} d x - \int 367416 x^{5} d x + \int 183708 x^{6} d x - \int 52488 x^{7} d x + \int 6561 x^{8} d x

\int 6561 d x = 6561 x

\int 52488 x d x = 26244 x^{2}

\int 183708 x^{2} d x = 61236 x^{3}

\int 367416 x^{3} d x = 91854 x^{4}

\int 459270 x^{4} d x = 91854 x^{5}

\int 367416 x^{5} d x = 61236 x^{6}

\int 183708 x^{6} d x = 26244 x^{7}

\int 52488 x^{7} d x = 6561 x^{8}

\int 6561 x^{8} d x = 729 x^{9}

= 6561 x - 26244 x^{2} + 61236 x^{3} - 91854 x^{4} + 91854 x^{5} - 61236 x^{6} + 26244 x^{7} - 6561 x^{8} + 729 x^{9}

解答补常数

= 6561 x - 26244 x^{2} + 61236 x^{3} - 91854 x^{4} + 91854 x^{5} - 61236 x^{6} + 26244 x^{7} - 6561 x^{8} + 729 x^{9} + C

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