What is the integral of (1-2x)/(sqrt(x-x^2)) ?

The integral of (1-2x)/(sqrt(x-x^2)) is 2sqrt(x(-x+1))+C

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

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

2 x (- x + 1) + C

Solution steps

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

Expand

\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

Apply the Sum Rule:

\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))

Simplify

= 2 x (- x + 1)

Add a constant to the solution

= 2 x (- x + 1) + C

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What is the integral of (1-2x)/(sqrt(x-x^2)) ?
The integral of (1-2x)/(sqrt(x-x^2)) is 2sqrt(x(-x+1))+C