$$\eqalign{ & \int {\arcsin xdx} = x\arcsin x - \int {xd\left( {\arcsin x} \right)} \cr & \int {xd\left( {\arcsin x} \right)} = \int {\frac{x}{{\sqrt {1 - {x^2}} }}dx} = \frac{1}{2}\int {\frac{{2x}}{{\sqrt {1 - {x^2}} }}dx} = \frac{1}{2}\int {\frac{1}{{\sqrt {1 - {x^2}} }}d\left( {{x^2}} \right)} = \cr & - \frac{1}{2}\int {\frac{1}{{\sqrt {1 - {x^2}} }}d\left( {1 - {x^2}} \right)} = - \frac{1}{2}\int {{{\left( {1 - {x^2}} \right)}^{ - \frac{1}{2}}}d\left( {1 - {x^2}} \right)} = - {\left( {1 - {x^2}} \right)^{\frac{1}{2}}} + C = \cr & - \sqrt {1 - {x^2}} + C \cr & \int {\arcsin xdx} = x\arcsin x + \sqrt {1 - {x^2}} + C \cr} $$

$%x\arcsin x + \sqrt {1 - {x^2}} + C$%