integral of sqrt(x^3-2x^2+x)

解答

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

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

求解步骤

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

化简

x^{3} - 2 x^{2} + x : x (x - 1)^{2}

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

(x - 1)^{2} = ((x - 1)),

假定

(x - 1) \geq 0

= \int x (x - 1) d x

乘开

x (x - 1) : x^{\frac{3}{2}} - x

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

\int x^{\frac{3}{2}} d x = \frac{2}{5} x^{\frac{5}{2}}

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

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

解答补常数

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

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