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

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

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

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

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

Solution steps

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

Expand

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

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

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

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

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

Add a constant to the solution

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

\int \frac{1}{3 x - x ^{2}} d x

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

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

\int (100 + x)^{4} d x

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