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

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

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

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

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

Solution steps

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

\int 1 d x = x

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

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

Add a constant to the solution

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

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