posted by MS .
Find arc length of y=logx from x=1 to x=2.
arc length=Int of [sqrt(1+1/x^2)]dx
=Int of [sqrt(1+x^2)/x^2]
=Int of [sqrt(1+x^2)]/x from x=1 to x=2.
How to proceed further to integrate?
x = tan u
dx = sec^2 u du
1+x^2 = sec^2 u
and you will wind up with some nice integrands involving tan u and sec u
I reached upto Int csc u sec^2 u du from u=arctan 1 to 2 but am not clear how to go further.
take a trip on over to wolframalpha and enter
integral sqrt(x+1)/x dx
and then click on the "Show Step-by-Step Solution" button (you may have to register first)
and it will show all the intricacies of the substitution.
Or, recall that sec^2 = 1+tan^2.