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

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

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

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

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

Solution steps

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

Multiply by the conjugate of

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

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

Simplify

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

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

\int 1 d x = x

\int - x^{2} + 1 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

\int \frac{1}{1 + cos ( x ) - sin ( x )} d x

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

\int x 8 - 2 x^{2} d x

\int \frac{- 2 x + 11}{x ^{2} - 8 x + 25} d x

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

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