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

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

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

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

arcsin (x) + 2 1 - x^{2} + C

Solution steps

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

Expand

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

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

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

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

= arcsin (x) - (- 2 1 - x^{2})

Simplify

= arcsin (x) + 2 1 - x^{2}

Add a constant to the solution

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

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