integral of (x-x^2)^2

解答

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

\frac{x ^{3}}{3} - \frac{x ^{4}}{2} + \frac{x ^{5}}{5} + C

求解步骤

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

乘开

(x - x^{2})^{2} : x^{2} - 2 x^{3} + x^{4}

= \int x^{2} - 2 x^{3} + x^{4} 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^{2} d x - \int 2 x^{3} d x + \int x^{4} d x

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

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

\int x^{4} d x = \frac{x ^{5}}{5}

= \frac{x ^{3}}{3} - \frac{x ^{4}}{2} + \frac{x ^{5}}{5}

解答补常数

= \frac{x ^{3}}{3} - \frac{x ^{4}}{2} + \frac{x ^{5}}{5} + C

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