The integral $\begin{align*} \int_0^{\pi} \sqrt{1 + 4 \sin^2 \frac{x}{2} - 4 \sin \frac{x}{2} \; dx} \end{align*}$ equals

Question

( A ) $4\sqrt{3} - 4 - \frac{\pi}{3}$
( B ) $\frac{2\pi}{3} - 4 - 4\sqrt{3}$
( C ) $4\sqrt{3} - 4$
( D ) $\pi - 4$