integral of (x(x-2))/((x-1)^2)

解答

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

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

求解步骤

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

使用分布积分法

= \frac{x ^{2} ( x - 2 )}{( x - 1 ) ^{2}} - \int \frac{2 x}{( x - 1 ) ^{3}} d x

\int \frac{2 x}{( x - 1 ) ^{3}} d x = - \frac{2}{x - 1} - \frac{1}{( x - 1 ) ^{2}}

= \frac{x ^{2} ( x - 2 )}{( x - 1 ) ^{2}} - (- \frac{2}{x - 1} - \frac{1}{( x - 1 ) ^{2}})

化简

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

解答补常数

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

p a r t ia l f r a c t i o n \frac{- 2 x ^{2}}{x ^{2} - 1}

\frac{d}{d x} (- 3 x^{2} + 24 x + 10)

\frac{\partial}{\partial y} (e^{x} sec^{2} (y))

t an g e n t f (x) = x^{2} - 8, (4, 8)

t an g e n t y = 9 x - 6 x, (1, 3)