Length of a Curve

The length of any curve is defined by $L= \int_{a'}^{b'}\sqrt{(f'(x))^2+1} \ dx$ .

In our problem $a'= \pi/3 \ and \ b'= \pi$

and $f(x)= 2 ln(sin (x/2)) \implies f'(x)= \frac{2}{sin(x/2)}\times (cos(x/2)\times 1/2$ );

$f'(x)=cot(x/2) \implies f'^2(x)= cot^2(x/2) \implies f'^2(x)+1= cot^2(x/2)+1= csc^2(x/2)$ ;

The length is now: $L= \int_{a'}^{b'} \sqrt {csc^2(x/2)} \ dx; where \ a'= \pi/3 \ and \ b'= \pi$ ;

$L= \int_{\pi/3}^{\pi} csc(x/2) dx = -2ln|cot(x/4)| \large |_{\pi/3}^{\pi}$ ;

$L= -2(ln|cot(\pi/12)|+ln|cot(\pi/4)|)= -2 ln|2-\sqrt{3}|$ ; $a=2 \ and \ b= 3 \implies \boxed{a+b=5}$