integral of x(1-2x)^5

解答

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

\frac{x ^{2}}{2} - \frac{10 x ^{3}}{3} + 10 x^{4} - 16 x^{5} + \frac{40 x ^{6}}{3} - \frac{32 x ^{7}}{7} + C

求解步骤

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

乘开

x (1 - 2 x)^{5} : x - 10 x^{2} + 40 x^{3} - 80 x^{4} + 80 x^{5} - 32 x^{6}

= \int x - 10 x^{2} + 40 x^{3} - 80 x^{4} + 80 x^{5} - 32 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 x d x - \int 10 x^{2} d x + \int 40 x^{3} d x - \int 80 x^{4} d x + \int 80 x^{5} d x - \int 32 x^{6} d x

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

\int 10 x^{2} d x = \frac{10 x ^{3}}{3}

\int 40 x^{3} d x = 10 x^{4}

\int 80 x^{4} d x = 16 x^{5}

\int 80 x^{5} d x = \frac{40 x ^{6}}{3}

\int 32 x^{6} d x = \frac{32 x ^{7}}{7}

= \frac{x ^{2}}{2} - \frac{10 x ^{3}}{3} + 10 x^{4} - 16 x^{5} + \frac{40 x ^{6}}{3} - \frac{32 x ^{7}}{7}

解答补常数

= \frac{x ^{2}}{2} - \frac{10 x ^{3}}{3} + 10 x^{4} - 16 x^{5} + \frac{40 x ^{6}}{3} - \frac{32 x ^{7}}{7} + C

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