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

解答

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

- \frac{1}{x} - 2 ln ∣ x ∣ + x + C

求解步骤

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

乘开

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

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

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

\int \frac{2}{x} d x = 2 ln ∣ x ∣

\int 1 d x = x

= - \frac{1}{x} - 2 ln ∣ x ∣ + x

解答补常数

= - \frac{1}{x} - 2 ln ∣ x ∣ + x + C

x^{2} + y^{2} + 2 x y y^{^{'}} = 0

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

d er i v a t i v e f (x) = \frac{x ^{2} - 2 x - 3}{x}

\frac{\partial}{\partial x} (3 x^{2} - 12 y)

d er i v a t i v e \frac{4}{x - 4} + \frac{27}{2 x + 2}