integral of (2-5x)^7

解答

\int (2 - 5 x)^{7} d x

128 x - 1120 x^{2} + 5600 x^{3} - 17500 x^{4} + 35000 x^{5} - 43750 x^{6} + 31250 x^{7} - \frac{78125 x ^{8}}{8} + C

求解步骤

\int (2 - 5 x)^{7} d x

乘开

(2 - 5 x)^{7} : 128 - 2240 x + 16800 x^{2} - 70000 x^{3} + 175000 x^{4} - 262500 x^{5} + 218750 x^{6} - 78125 x^{7}

= \int 128 - 2240 x + 16800 x^{2} - 70000 x^{3} + 175000 x^{4} - 262500 x^{5} + 218750 x^{6} - 78125 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 2240 x d x + \int 16800 x^{2} d x - \int 70000 x^{3} d x + \int 175000 x^{4} d x - \int 262500 x^{5} d x + \int 218750 x^{6} d x - \int 78125 x^{7} d x

\int 128 d x = 128 x

\int 2240 x d x = 1120 x^{2}

\int 16800 x^{2} d x = 5600 x^{3}

\int 70000 x^{3} d x = 17500 x^{4}

\int 175000 x^{4} d x = 35000 x^{5}

\int 262500 x^{5} d x = 43750 x^{6}

\int 218750 x^{6} d x = 31250 x^{7}

\int 78125 x^{7} d x = \frac{78125 x ^{8}}{8}

= 128 x - 1120 x^{2} + 5600 x^{3} - 17500 x^{4} + 35000 x^{5} - 43750 x^{6} + 31250 x^{7} - \frac{78125 x ^{8}}{8}

解答补常数

= 128 x - 1120 x^{2} + 5600 x^{3} - 17500 x^{4} + 35000 x^{5} - 43750 x^{6} + 31250 x^{7} - \frac{78125 x ^{8}}{8} + C

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