$($a$)$ Express $\frac{x+1}{\sqrt[\phantom{2}]{1-x^2}}$ in the form of

$\frac{A\frac{d}{dx}(1-x^2)+B}{\sqrt[\phantom{2}]{1-x^2}}$

where A and $B$ are constants. Hence, evaluate

${\displaystyle \underset{0}{\overset{\frac{1}{2}}{}}}\frac{x+1}{\sqrt[\phantom{2}]{1-x^2}}dx$