integral of (2-3x)^7

解答

\int (2 - 3 x)^{7} d x

128 x - 672 x^{2} + 2016 x^{3} - 3780 x^{4} + 4536 x^{5} - 3402 x^{6} + 1458 x^{7} - \frac{2187 x ^{8}}{8} + C

求解步骤

\int (2 - 3 x)^{7} d x

乘开

(2 - 3 x)^{7} : 128 - 1344 x + 6048 x^{2} - 15120 x^{3} + 22680 x^{4} - 20412 x^{5} + 10206 x^{6} - 2187 x^{7}

= \int 128 - 1344 x + 6048 x^{2} - 15120 x^{3} + 22680 x^{4} - 20412 x^{5} + 10206 x^{6} - 2187 x^{7} 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 128 d x - \int 1344 x d x + \int 6048 x^{2} d x - \int 15120 x^{3} d x + \int 22680 x^{4} d x - \int 20412 x^{5} d x + \int 10206 x^{6} d x - \int 2187 x^{7} d x

\int 128 d x = 128 x

\int 1344 x d x = 672 x^{2}

\int 6048 x^{2} d x = 2016 x^{3}

\int 15120 x^{3} d x = 3780 x^{4}

\int 22680 x^{4} d x = 4536 x^{5}

\int 20412 x^{5} d x = 3402 x^{6}

\int 10206 x^{6} d x = 1458 x^{7}

\int 2187 x^{7} d x = \frac{2187 x ^{8}}{8}

= 128 x - 672 x^{2} + 2016 x^{3} - 3780 x^{4} + 4536 x^{5} - 3402 x^{6} + 1458 x^{7} - \frac{2187 x ^{8}}{8}

解答补常数

= 128 x - 672 x^{2} + 2016 x^{3} - 3780 x^{4} + 4536 x^{5} - 3402 x^{6} + 1458 x^{7} - \frac{2187 x ^{8}}{8} + C

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