Find the arc length of

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

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  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.

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