How should I try to evaluate this integral $\int_0^\pi \sqrt{1+4\sin^2\frac x2 - 4\sin\frac x2}\;dx$? Suppose that we are given the following integral:
$$\int_0^\pi \sqrt{1+4\sin^2\frac x2 - 4\sin\frac x2}\;dx.$$
(Original screenshot)
And the answer is one of these :-  

  
*
  
*$4\sqrt3-4-\frac\pi3$
  
*$\pi-4$
  
*$\frac{2\pi}3 - 4 - 4\sqrt3$
  
*$4\sqrt3-4$
  

(original screenshot  )
Is there a way to check each answer and reach to the question (like in case of indefinite integral I can differentiate the options and match it with question and tick the correct answer) or maybe confirm that this value is correct for the question without solving.

\begin{align}
\int_0^\pi\sqrt{4\sin^2\frac{x}{2}-4\sin\frac{x}{2}+1}\,dx&=\int_0^\pi\sqrt{\left(2\sin\frac{x}{2}-1\right)^2}\,dx\\
&=\int_0^\frac{\pi}{3}\left(1-2\sin\frac{x}{2}\right)\,dx+\int_\frac{\pi}{3}^\pi\left(2\sin\frac{x}{2}-1\right)\,dx\\
&=\left[x+4\cos\frac{x}{2}\right]_0^\frac{\pi}{3}+\left[-4\cos\frac{x}{2}-x\right]_\frac{\pi}{3}^\pi\\
&=\frac{\pi}{3}+4\cos\frac{\pi}{6}-4-\pi+4\cos\frac{\pi}{6}+\frac{\pi}{3}\\
&=4\sqrt{3}-4-\frac{\pi}{3}
\end{align}
If this is multiple choices, I'd immediately answer 1 because its value greater than $0$ and we can easily notice that the answer must have term of $\pi$ because there is term integral of $1$ that yield $x$.
