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

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

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

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

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

Solution steps

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

Expand

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

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

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

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

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

Simplify

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

Add a constant to the solution

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

x \to \infty lim (\frac{x}{- 1})

6 e^{3 x} (3 cos (4 x) - 4 sin (4 x))

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

y = x^{2}, (3, 9)

x \to - 2 lim (2 x^{2} - x + 3)

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