Analytical Integration of Arcsin x

We can integrate  
\[f(x)=sin^{-1} x\]
  by parts.
Let  
\[u=sin^{-1}x\]
  and  
\[v'=1\]
.
Then  
\[u'=\frac{1}{\sqrt{1-x^2}}\]
,  
\[v=x\]
.
Integration by parts now gives
\[\begin{equation} \begin{aligned} \int 1 \times sin^{-1}x dx &= xsin^{-1}x - \int x \times \frac{1}{\sqrt{1-x^2}} dx \\ &= xsin^{-1}x +(1-x^2) +c \end{aligned} \end{equation}\]