Well, you could also have started with a straight forward substitution: ##y=\arcsin x##
This means that ##x=\sin y##.
You're rid of that pesky arcsin and get easier functions to integrate.
$$\int (\arcsin x)^2 dx = \int y^2 d(\sin y)$$
Now do integration by parts:
$$\int y^2 d(\sin y) = y^2 \sin y - \int 2y \cos y dy$$
Repeat integration by parts, and after that back substitute x for y...
Sorry I just looked back at this thread, this is interesting, but I don't know how to follow it. Could someone explain?