integral of x(5-x)^5

解答

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

\frac{3125 x ^{2}}{2} - \frac{3125 x ^{3}}{3} + \frac{625 x ^{4}}{2} - 50 x^{5} + \frac{25 x ^{6}}{6} - \frac{x ^{7}}{7} + C

求解步骤

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

乘开

x (5 - x)^{5} : 3125 x - 3125 x^{2} + 1250 x^{3} - 250 x^{4} + 25 x^{5} - x^{6}

= \int 3125 x - 3125 x^{2} + 1250 x^{3} - 250 x^{4} + 25 x^{5} - x^{6} 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 3125 x d x - \int 3125 x^{2} d x + \int 1250 x^{3} d x - \int 250 x^{4} d x + \int 25 x^{5} d x - \int x^{6} d x

\int 3125 x d x = \frac{3125 x ^{2}}{2}

\int 3125 x^{2} d x = \frac{3125 x ^{3}}{3}

\int 1250 x^{3} d x = \frac{625 x ^{4}}{2}

\int 250 x^{4} d x = 50 x^{5}

\int 25 x^{5} d x = \frac{25 x ^{6}}{6}

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

= \frac{3125 x ^{2}}{2} - \frac{3125 x ^{3}}{3} + \frac{625 x ^{4}}{2} - 50 x^{5} + \frac{25 x ^{6}}{6} - \frac{x ^{7}}{7}

解答补常数

= \frac{3125 x ^{2}}{2} - \frac{3125 x ^{3}}{3} + \frac{625 x ^{4}}{2} - 50 x^{5} + \frac{25 x ^{6}}{6} - \frac{x ^{7}}{7} + C

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