The integral \int_0^\pi \sqrt{1+4\sin^2(\frac{x}{2})}\sin(\frac{x}{2})dx equals

I=\int_0^\pi \sqrt{1+4\sin^2(x/2)}\sin(x/2)dx
I=\int_0^\pi |2\sin(x/2)|dx
If 0<x<\frac{\pi}{3}, then 2\sin(x/2)<0
And if \frac{\pi}{3}<x<0, then 2\sin(x/2)>0
\therefore I=\int_0^{\pi/3} -(2\sin(x/2))dx+\int_{\pi}^{\pi/3} (2\sin(x/2))dx
I=[4\cos(x/2)+x]0^{\pi/3}+[-4\cos(x/2)-x]{\pi}^{\pi/3}
I=4\frac{\sqrt{3}}{2}+\frac{\pi}{3}-(4+0)+(-4\frac{\sqrt{3}}{2}-\frac{\pi}{3}+0+\pi)
I=4\sqrt{3}-\frac{\pi}{3}