arc length of logarithmic curve

The arc lengthMathworldPlanetmath of the graph of logarithm function ( is expressible in closed form (other cases are listed in the entry arc length of parabola).  The usual arc length


gives, if  0<a<b,  for  f(x):=lnx,  f(x)=1x,  the expression

s=ab1+x2x𝑑x. (1)

Here, finding a suitable substitution for integration may be a bit difficult.  E.g.  x:=tant leads to


the substitution  x:=sinht  to


which both seem to require a new substitution.  As well the Euler’s substitutions (1st and 2nd ones) lead to awkward rational functions as integrands.

But there is the straightforward substitution




(see area functions) and then


Using this antiderivative, one can obtain the arc length (1).  For example, if  a=3  and  b=15,  the result is  s=2+ln35.

As for finding the arc length of the graph of the functionDlmfDlmfMathworldPlanetmathxex,  which actually is the same curve as the graph of the inverse functionxlnx,  one may write the expression

s=αβ1+e2x𝑑x. (2)

Since here the substitution


shows that


we see that it’s really a question of the same task as above.  The antiderivative is

Title arc length of logarithmic curve
Canonical name ArcLengthOfLogarithmicCurve
Date of creation 2013-03-22 19:01:45
Last modified on 2013-03-22 19:01:45
Owner pahio (2872)
Last modified by pahio (2872)
Numerical id 11
Author pahio (2872)
Entry type Example
Classification msc 53A04
Classification msc 26A42
Classification msc 26A09
Classification msc 26A06
Synonym arc length of exponential curve