Question

Find the length of the curve, l. y = ln(cos x), 0 ≤ x ≤ π/3

Answer 1

`y=\ln\cos x\implies(\mathrm dy)/(\mathrm dx)=-(\sin x)/(\cos x)=-\tan x`


`\displaystyle\int_(\ell)\mathrm dS=\int_(x=0)^(x=\pi/3)√(1+(-\tan x)^2)\,\mathrm dx=\int_0^(\pi/3)√(\sec^2x)\,\mathrm dx`

Recall that
`√(x^2)=|x|`, but since
`\sec x>0` over the integration domain, we can reduce
`|\sec x|=\sec x`. So we have


`\displaystyle\int_0^(\pi/3)\sec x\,\mathrm dx=\ln|\sec x+\tan x|\bigg|_(x=0)^(x=\pi/3)=\ln\left|\sec\frac\pi3+\tan\frac\pi3\right|=\ln(2+\sqrt3)`