Example 2.56. Calculate the length of the arc of the curve $x=\ln (\cos y)$ between the points $y=0$ and $y=\frac{\pi}{3}$.
Solution. We will use formula (2.25).

\[
\begin{gathered}
L=\int_{0}^{\pi / 3} \sqrt{1+\left(-\frac{\sin y}{\cos y}\right)^{2}} d y=\int_{0}^{\pi / 3} \sqrt{1+\operatorname{tg}^{2} y} d y=\int_{0}^{\pi / 3} \frac{d y}{\cos y}= \\
=\left.\ln \left|\operatorname{tg}\left(\frac{\pi}{4}+\frac{y}{2}\right)\right|\right|_{0} ^{\pi / 3}=\ln \operatorname{tg}\left(\frac{5 \pi}{12}\right)-\ln \left(\frac{\pi}{4}\right)
\end{gathered}
\]