The functions $$x \;\mapsto\; \ln\sin{x} \quad \mbox{and} \quad x \;\mapsto\, \ln\cos{x}$$ belong to the few real functions, the arc length of which are expressible in closed form (other ones are mentionned in the entry arc length of parabola).

We calculate the arc length of the curve $$y \;=\; \ln\sin{x}\qquad (0 \;<\; a \;\leqq\; x \;\leqq\; \frac{\pi}{2}).$$ By the chain rule, we have $$y' \;=\; \frac{1}{\sin{x}}\cdot\cos{x} \;=\; \cot{x}.$$ Hence the arc length is $$s \;=\; \int_a^{\frac{\pi}{2}}\!\sqrt{1+(\cot{x})^2}\,dx \;=\; \int_a^{\frac{\pi}{2}}\!\frac{1}{\sin{x}}\,dx \;=\; \sijoitus{a}{\quad\frac{\pi}{2}}\!\ln|\tan\frac{x}{2}| \;=\;\ln1-\ln|\tan\frac{a}{2}| \;=\; \ln\cot\frac{a}{2}$$ (see integration of rational function of sine and cosine).