Show that the length of the curve y= log Secx between the points x = 0 and x = π/3 Is log(2+ √3 )

Question

To find the length of the curve.

Answer 1

y=log(sec(x)) so dy/dx=1/sec(x).sec(x)tan(x)=tan(x).

Consider a small line segment of th curve ds=sqrt(dy^2+dx^2) where dy and dx are small displacements from a point P(x,y) on the curve. ds/dx=sqrt((dy/dx)^2+1)=sqrt(tan^2(x)+1)=sec(x) and ds=sec(x)dx. So if S[0,(pi)/3](sec(x)dx) represents the integral for 0<x<(pi)/3, then the length of the curve is the integral (sum of the segments ds).

Multiply the integrand by (sec(x)+tan(x))/(sec(x)+tan(x)): (sec^2(x)+sec(x)tan(x))/(sec(x)+tan(x))dx. The top is the differential of the bottom, so the integral is ln|sec(x)+tan(x)|[0,(pi)/3]=ln|2+sqrt(3)|-ln|1|=ln(2+sqrt(3)).