$\int \sqrt {-(4-4x+x^2)+4}.dx$
$\int \sqrt {(2)^2 -(x-2)^2 }.dx$
$(x-2)=t=> dx=dt$
=>$\int \sqrt {(2)^2 -(t)^2 }.dt$
=> $\large\frac{t}{2} . $$\sqrt {(2)^2-t^2}+\large\frac{4}{2}.$$ \sin ^{-1} \bigg(\large\frac{t}{2}\bigg)$$+c$
$\large\frac{(x-2)}{2}$$\sqrt {4x-x^2}+2 \sin ^{-1} \bigg(\frac{x-2}{2}\bigg)+c $
Hence b is the correct answer.