Find arc length for  y=log x for x=1 to x=2.
dy/dx=1/x, (dy/dx)2=1/x2 , hence arc length=Int of sqrt(1+1/x2)dx from x=1 to 2
= Int of [sqrt(1+x^2)]dx/x=Int [sqrt(1+x^2)sqrt(1+x^2)dx]/x (sqrt(1+x^2) by rationalizing the numerator.
= Int (1+x2)dx/x (sqrt(1+x^2).
Please advise how to proceed further to integrate?

You can consider a trigonometric substitution, ie x=tanθ

Int of [sqrt(1+x^2)]dx/x  from x=1 to 2 becomes

Int of [sqrt(1+tan^2 θ)]/tanθ * sec^2 θ dθ   from θ= pi/4 to θ = arctan(2)

= Int of sec θ /tanθ * sec^2 θ dθ   from θ= pi/4 to θ = arctan(2)

= Int of sec^3  θ /tanθ  dθ   from θ= pi/4 to θ = arctan(2)

Hopefully I have given you sufficient assistance and you are able to continue from here. Peace.  


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Frederick Koh


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