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

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

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

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

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

Solution steps

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

Expand

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

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

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

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

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

Simplify

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

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

Add a constant to the solution

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

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