calculus help?

Question

Verify that the length of arc of the graph of f(x)=ln |coth x + csch x| from the pt. where x=ln2 to the point where x=ln4 is ln(5/2)

Answer 1

df/dx = [1/ {cschx + coth x}*{-csch^2 x - cschx coth x}
df/dx = - cschx *[{cschx + coth x}] / {cschx + coth x}

df/dx = - csch x

length = ∫ sqrt (1+csch^2 x) dx

length = ∫ coth x dx

= [log sinh x ] in limits
= log [e^x - e^-x)/2]
= log [e^log4 - e^-log4] - log [e^log2 - e^-log2]

= log [4 - 1/4] - log [2 - 1/2]
= log [15/4] - log [3/2]
= log [15/4*2/3] = log (5/2)