integral of x(1-5x^2)^6

解答

\int x (1 - 5 x^{2})^{6} d x

\frac{x ^{2}}{2} - \frac{15 x ^{4}}{2} + \frac{125 x ^{6}}{2} - \frac{625 x ^{8}}{2} + \frac{1875 x ^{10}}{2} - \frac{3125 x ^{12}}{2} + \frac{15625 x ^{14}}{14} + C

求解步骤

\int x (1 - 5 x^{2})^{6} d x

乘开

x (1 - 5 x^{2})^{6} : x - 30 x^{3} + 375 x^{5} - 2500 x^{7} + 9375 x^{9} - 18750 x^{11} + 15625 x^{13}

= \int x - 30 x^{3} + 375 x^{5} - 2500 x^{7} + 9375 x^{9} - 18750 x^{11} + 15625 x^{13} 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 x d x - \int 30 x^{3} d x + \int 375 x^{5} d x - \int 2500 x^{7} d x + \int 9375 x^{9} d x - \int 18750 x^{11} d x + \int 15625 x^{13} d x

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

\int 30 x^{3} d x = \frac{15 x ^{4}}{2}

\int 375 x^{5} d x = \frac{125 x ^{6}}{2}

\int 2500 x^{7} d x = \frac{625 x ^{8}}{2}

\int 9375 x^{9} d x = \frac{1875 x ^{10}}{2}

\int 18750 x^{11} d x = \frac{3125 x ^{12}}{2}

\int 15625 x^{13} d x = \frac{15625 x ^{14}}{14}

= \frac{x ^{2}}{2} - \frac{15 x ^{4}}{2} + \frac{125 x ^{6}}{2} - \frac{625 x ^{8}}{2} + \frac{1875 x ^{10}}{2} - \frac{3125 x ^{12}}{2} + \frac{15625 x ^{14}}{14}

解答补常数

= \frac{x ^{2}}{2} - \frac{15 x ^{4}}{2} + \frac{125 x ^{6}}{2} - \frac{625 x ^{8}}{2} + \frac{1875 x ^{10}}{2} - \frac{3125 x ^{12}}{2} + \frac{15625 x ^{14}}{14} + C

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