integral of ((x^a-x^b)^2)/(sqrt(x))

解答

\int \frac{( x ^{a} - x ^{b} ) ^{2}}{x} d x

\frac{2 x ^{\frac{4 a + 1}{2}}}{4 a + 1} - \frac{4 x ^{\frac{2 a + 2 b + 1}{2}}}{2 a + 2 b + 1} + \frac{2 x ^{\frac{4 b + 1}{2}}}{4 b + 1} + C

求解步骤

\int \frac{( x ^{a} - x ^{b} ) ^{2}}{x} d x

乘开

\frac{( x ^{a} - x ^{b} ) ^{2}}{x} : x^{\frac{4 a - 1}{2}} - 2 x^{\frac{2 a + 2 b - 1}{2}} + x^{\frac{4 b - 1}{2}}

= \int x^{\frac{4 a - 1}{2}} - 2 x^{\frac{2 a + 2 b - 1}{2}} + x^{\frac{4 b - 1}{2}} 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{4 a - 1}{2}} d x - \int 2 x^{\frac{2 a + 2 b - 1}{2}} d x + \int x^{\frac{4 b - 1}{2}} d x

\int x^{\frac{4 a - 1}{2}} d x = \frac{2 x ^{\frac{4 a + 1}{2}}}{4 a + 1}

\int 2 x^{\frac{2 a + 2 b - 1}{2}} d x = \frac{4 x ^{\frac{2 a + 2 b + 1}{2}}}{2 a + 2 b + 1}

\int x^{\frac{4 b - 1}{2}} d x = \frac{2 x ^{\frac{4 b + 1}{2}}}{4 b + 1}

= \frac{2 x ^{\frac{4 a + 1}{2}}}{4 a + 1} - \frac{4 x ^{\frac{2 a + 2 b + 1}{2}}}{2 a + 2 b + 1} + \frac{2 x ^{\frac{4 b + 1}{2}}}{4 b + 1}

解答补常数

= \frac{2 x ^{\frac{4 a + 1}{2}}}{4 a + 1} - \frac{4 x ^{\frac{2 a + 2 b + 1}{2}}}{2 a + 2 b + 1} + \frac{2 x ^{\frac{4 b + 1}{2}}}{4 b + 1} + C

d er i v a t i v e \frac{2}{t - 3}

s l o p e (- 2.7) (4.3)

3 (y^{4} + 1) d x + 4 x y^{3} d y = 0

\frac{d y}{d x} = \frac{2 x y}{1 + x ^{2}}

in v erse l a pl a ce \frac{s}{s ^{2} - k ^{2}}