Arc Length of y=lncosx from 0 to pi/4

Since y = ln(cos x), differentiating yields dy/dx = (1/cos x) * -sin x = -tan x.

So, √(1 + (dy/dx)^2)
= √(1 + tan^2(x))
= √(sec^2(x))
= sec x, since sec x > 0 for x in [0, π/4].

Therefore, the arc length equals
∫(x = 0 to π/4) sec x dx
= ln |sec x + tan x| {for x = 0 to π/4}
= ln(√2 + 1).

I hope this helps!