Looking to integrate arcsin(x)*dx from 0 to 1 by method of int by parts. You get an improper integral when doing this, but want to do it regardless.
$\displaystyle u = arcsinx \implies du = \frac{1}{\sqrt{1-x^2}} dx$
and,
$\displaystyle dv = dx \implies v = x$
using integration by parts, the expression is
$\displaystyle uv - \int v. \text{du}$
$\displaystyle = x \times arcsin(x) - \int \frac{x}{\sqrt{1-x^2}} dx$
Now for $\displaystyle \int \frac{x}{\sqrt{1-x^2}} dx$, use substitution:
$\displaystyle u = 1-x^2$
and integrate