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

解

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

2 x (- x + 1) + C

解答ステップ

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

拡張

\frac{1 - 2 x}{x - x ^{2}} : \frac{1}{x - x ^{2}} - \frac{2 x}{x - x ^{2}}

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

総和規則を適用する:

\int f (x) \pm g (x) d x = \int f (x) d x \pm \int g (x) d x

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

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

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

= arcsin (2 x - 1) - (- 2 x (- x + 1) + arcsin (2 x - 1))

簡素化

= 2 x (- x + 1)

定数を解答に追加する

= 2 x (- x + 1) + C