arc length of logarithmic curve

The arc length of the graph of logarithm function (http://planetmath.org/NaturalLogarithm2) is expressible in closed form (other cases are listed in the entry arc length of parabola).  The usual arc length

$$s={\int }_a^b\sqrt{1+{(f^{\prime }(x))}^2}𝑑x$$

gives, if  ,  for  $f(x):=\ln x$,  $f^{\prime }(x)=\frac{1}{x}$,  the expression

$s={\displaystyle {\int }_a^b}{\displaystyle \frac{\sqrt{1+x^2}}{x}}𝑑x.$ (1)

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

$$\int \frac{\sqrt{1+x^2}}{x}𝑑x=\int \frac{dt}{\sin t{\cos }^{2}t},$$

the substitution  $x:=\sinh t$  to

$$\int \frac{\sqrt{1+x^2}}{x}𝑑x=\int \frac{{\cosh }^{2}t}{\sinh t}𝑑t,$$

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

$$\sqrt{1+x^2}:=t,x=\sqrt{t^2-1},dx=\frac{tdt}{\sqrt{t^2-1}}$$

yielding

$$\int \frac{\sqrt{1+x^2}}{x}𝑑x=\int \frac{t^{2}dt}{t^2-1}=t+\frac{1}{2}\ln \frac{t-1}{t+1}+C=t-\mathrm{arcoth}t+C$$

(see area functions) and then

$$\int \frac{\sqrt{1+x^2}}{x}𝑑x=\sqrt{1+x^2}+\frac{1}{2}\ln \frac{\sqrt{1+x^2}-1}{\sqrt{1+x^2}+1}+C=\sqrt{1+x^2}+\ln \frac{x}{1+\sqrt{1+x^2}}+C.$$

Using this antiderivative, one can obtain the arc length (1).  For example, if  $a=\sqrt{3}$  and  $b=\sqrt{15}$,  the result is  $s=2+\ln \frac{3}{\sqrt{5}}$.

As for finding the arc length of the graph of the http://planetmath.org/node/2541exponential function  $x\mapsto e^x$,  which actually is the same curve as the graph of the inverse function  $x\mapsto \ln x$,  one may write the expression

$s={\displaystyle {\int }_{\alpha }^{\beta }}\sqrt{1+e^{2x}}𝑑x.$ (2)

Since here the substitution

$$e^x:=t,x=\ln t,dx=\frac{dt}{t}$$

shows that

$$\int \sqrt{1+e^{2x}}𝑑x=\int \frac{\sqrt{1+t^2}}{t}𝑑t,$$

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

$$\int \sqrt{1+e^{2x}}𝑑x=\sqrt{1+e^{2x}}-\mathrm{arsinh}{e}^{-x}+C=\sqrt{1+e^{2x}}+\ln \frac{e^x}{1+\sqrt{1+e^{2x}}}+C.$$

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