Question

Determine la longitud de la curva Y=ln cos(x) en el intervalo (0 , \pi /3 ).

Answer 1

                           $\displaystyle L=\int\limits_{0}^{\pi/3}\sqrt{1+\left(\dfrac{-\sin x}{\cos x}\right)^2}~dx\\ \\ \\ L=\int\limits_{0}^{\pi/3}\sqrt{1+\tan^2x}~dx\\ \\ \\ L=\int\limits_{0}^{\pi/3}\sec{x}~dx\\ \\ \\ L=\left.\ln|\sec x+\tan x|\right|_{0}^{\pi/3}\\ \\ \\ L=\ln \left(2+\sqrt{3}\right)-\ln 1\\ \\ \\ \boxed{L=\ln \left(2+\sqrt{3}\right)}$