integral of sqrt(x/(2-x))

解答

\int \frac{x}{2 - x} d x

- 2 \frac{x}{2 - x} + \frac{x}{2 - x} x - 2 arcsin (\frac{1}{2} 2 - x) + C

求解步骤

\int \frac{x}{2 - x} d x

使用换元积分法

= \int - \frac{- u + 2}{u} d u

提出常数:

\int a \cdot f (x) d x = a \cdot \int f (x) d x

= - \int \frac{- u + 2}{u} d u

使用分布积分法

= - (\frac{- u + 2}{u} u - \int - \frac{1}{u ^{\frac{1}{2}} - u + 2} d u)

\int - \frac{1}{u ^{\frac{1}{2}} - u + 2} d u = - 2 arcsin (\frac{1}{2} u)

= - (\frac{- u + 2}{u} u - (- 2 arcsin (\frac{1}{2} u)))

u = 2 - x

代回

= - (\frac{- ( 2 - x ) + 2}{2 - x} (2 - x) - (- 2 arcsin (\frac{1}{2} 2 - x)))

化简

- (\frac{- ( 2 - x ) + 2}{2 - x} (2 - x) - (- 2 arcsin (\frac{1}{2} 2 - x))) : - 2 \frac{x}{2 - x} + \frac{x}{2 - x} x - 2 arcsin (\frac{1}{2} 2 - x)

= - 2 \frac{x}{2 - x} + \frac{x}{2 - x} x - 2 arcsin (\frac{1}{2} 2 - x)

解答补常数

= - 2 \frac{x}{2 - x} + \frac{x}{2 - x} x - 2 arcsin (\frac{1}{2} 2 - x) + C

\int lo g_{e} (x^{2}) d x

l a pl a ce t r an s f or m e^{5 t} (- 2 cos (t) + 12 sin (t))

\frac{\partial}{\partial x} (y 1 + x^{7})

\int \frac{11 x ^{2}}{( 144 + x ^{2} ) ^{2}} d x

\frac{d}{d x} (x^{6} ln (7 x))