integral of (5x-2)^9

解答

\int (5 x - 2)^{9} d x

\frac{390625 x ^{10}}{2} - 781250 x^{9} + 1406250 x^{8} - 1500000 x^{7} + 1050000 x^{6} - 504000 x^{5} + 168000 x^{4} - 38400 x^{3} + 5760 x^{2} - 512 x + C

求解步骤

\int (5 x - 2)^{9} d x

乘开

(5 x - 2)^{9} : 1953125 x^{9} - 7031250 x^{8} + 11250000 x^{7} - 10500000 x^{6} + 6300000 x^{5} - 2520000 x^{4} + 672000 x^{3} - 115200 x^{2} + 11520 x - 512

= \int 1953125 x^{9} - 7031250 x^{8} + 11250000 x^{7} - 10500000 x^{6} + 6300000 x^{5} - 2520000 x^{4} + 672000 x^{3} - 115200 x^{2} + 11520 x - 512 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 1953125 x^{9} d x - \int 7031250 x^{8} d x + \int 11250000 x^{7} d x - \int 10500000 x^{6} d x + \int 6300000 x^{5} d x - \int 2520000 x^{4} d x + \int 672000 x^{3} d x - \int 115200 x^{2} d x + \int 11520 x d x - \int 512 d x

\int 1953125 x^{9} d x = \frac{390625 x ^{10}}{2}

\int 7031250 x^{8} d x = 781250 x^{9}

\int 11250000 x^{7} d x = 1406250 x^{8}

\int 10500000 x^{6} d x = 1500000 x^{7}

\int 6300000 x^{5} d x = 1050000 x^{6}

\int 2520000 x^{4} d x = 504000 x^{5}

\int 672000 x^{3} d x = 168000 x^{4}

\int 115200 x^{2} d x = 38400 x^{3}

\int 11520 x d x = 5760 x^{2}

\int 512 d x = 512 x

= \frac{390625 x ^{10}}{2} - 781250 x^{9} + 1406250 x^{8} - 1500000 x^{7} + 1050000 x^{6} - 504000 x^{5} + 168000 x^{4} - 38400 x^{3} + 5760 x^{2} - 512 x

解答补常数

= \frac{390625 x ^{10}}{2} - 781250 x^{9} + 1406250 x^{8} - 1500000 x^{7} + 1050000 x^{6} - 504000 x^{5} + 168000 x^{4} - 38400 x^{3} + 5760 x^{2} - 512 x + C

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