Question

Find the arc length of

y = ln(cos x)
over x = [0, pi/4]

Answer 1

We have to integrate sqrt[1+y'2] from 0 to pi/4

1+y'^2 = 1+tan^2(x) = 1/cos^2(x)

So, the arc length is:

Integral from 0 to pi/4 of 1/cos(x)

= Integral from pi/4 to p1/2 of 1/sin(x)


We can write:

1/sin(x) =

1/[2sin(1/2 x) cos(1/2 x)] =

[sin^2(1/2x) + cos^2(1/2x)]
/[2sin(1/2 x) cos(1/2 x)] =


1/2 [tan(1/2 x) + cot(1/2 x)]

Which is trivial to integrate.