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From: J. Clarke on 14 May 2010 10:33
On 5/14/2010 9:53 AM, andrews wrote:
> I am not a student, thus it is not for a homework.
> With the answers I could find the solution.
> It is based on int(df/f) = lnf where f is any function
> Thanks for this

Just for future reference, Maxima (GPL computer algebra system) solved
this instantly. If you put it in the form "1/(x*ln(x))" then it's in
the Schaums mathematical handbook ($12.95 from Amazon) and the Ti-89
calculator solved it as well. I think you'll find having one or all of
those on hand will be a tremendous convenience if you run into this kind
of question often.




From: Greg Neill on 14 May 2010 11:15
J. Clarke wrote:
> On 5/14/2010 9:53 AM, andrews wrote:
>> I am not a student, thus it is not for a homework.
>> With the answers I could find the solution.
>> It is based on int(df/f) = lnf where f is any function
>> Thanks for this
>
> Just for future reference, Maxima (GPL computer algebra system) solved
> this instantly. If you put it in the form "1/(x*ln(x))" then it's in
> the Schaums mathematical handbook ($12.95 from Amazon) and the Ti-89
> calculator solved it as well. I think you'll find having one or all of
> those on hand will be a tremendous convenience if you run into this kind
> of question often.

Wolfram's online integrator is also handy.

Interestingly, it can resolve 1/(x*ln(x)) but
not 1/ln(x^x).

http://integrals.wolfram.com/index.jsp?expr=1%2F%28x*ln%28x%29%29&random=false


From: A N Niel on 14 May 2010 14:30
In article <AOdHn.100119$ae7.98318(a)unlimited.newshosting.com>, Greg
Neill <gneillRE(a)MOVEsympatico.ca> wrote:

> J. Clarke wrote:
> > On 5/14/2010 9:53 AM, andrews wrote:
> >> I am not a student, thus it is not for a homework.
> >> With the answers I could find the solution.
> >> It is based on int(df/f) = lnf where f is any function
> >> Thanks for this
> >
> > Just for future reference, Maxima (GPL computer algebra system) solved
> > this instantly. If you put it in the form "1/(x*ln(x))" then it's in
> > the Schaums mathematical handbook ($12.95 from Amazon) and the Ti-89
> > calculator solved it as well. I think you'll find having one or all of
> > those on hand will be a tremendous convenience if you run into this kind
> > of question often.
>
> Wolfram's online integrator is also handy.
>
> Interestingly, it can resolve 1/(x*ln(x)) but
> not 1/ln(x^x).
>
> http://integrals.wolfram.com/index.jsp?expr=1%2F%28x*ln%28x%29%29&random=false
>
>

So wolfram is smart, like Maple ... here is Maple:

int(1/x/log(x),x);
ln(ln(x))
int(1/log(x^x),x);
/ 1 \
int|------, x|
| / x\ |
\ln\x / /
int(1/log(x^x),x) assuming x>0;
ln(ln(x))
From: J. Clarke on 14 May 2010 14:40
On 5/14/2010 11:15 AM, Greg Neill wrote:
> J. Clarke wrote:
>> On 5/14/2010 9:53 AM, andrews wrote:
>>> I am not a student, thus it is not for a homework.
>>> With the answers I could find the solution.
>>> It is based on int(df/f) = lnf where f is any function
>>> Thanks for this
>>
>> Just for future reference, Maxima (GPL computer algebra system) solved
>> this instantly. If you put it in the form "1/(x*ln(x))" then it's in
>> the Schaums mathematical handbook ($12.95 from Amazon) and the Ti-89
>> calculator solved it as well. I think you'll find having one or all of
>> those on hand will be a tremendous convenience if you run into this kind
>> of question often.
>
> Wolfram's online integrator is also handy.
>
> Interestingly, it can resolve 1/(x*ln(x)) but
> not 1/ln(x^x).
>
> http://integrals.wolfram.com/index.jsp?expr=1%2F%28x*ln%28x%29%29&random=false

Hmm. Mathematica 7 the same. Wonder if anything else other than Maxima
(and presumably Macsyma) handles 1/ln(x^x) directly? Anybody out there
with Maple want to try it?
From: Robert Israel on 14 May 2010 15:46
A N Niel <anniel(a)nym.alias.net.invalid> writes:

> In article <AOdHn.100119$ae7.98318(a)unlimited.newshosting.com>, Greg
> Neill <gneillRE(a)MOVEsympatico.ca> wrote:
>
> > J. Clarke wrote:
> > > On 5/14/2010 9:53 AM, andrews wrote:
> > >> I am not a student, thus it is not for a homework.
> > >> With the answers I could find the solution.
> > >> It is based on int(df/f) = lnf where f is any function
> > >> Thanks for this
> > >
> > > Just for future reference, Maxima (GPL computer algebra system) solved
> > > this instantly. If you put it in the form "1/(x*ln(x))" then it's in
> > > the Schaums mathematical handbook ($12.95 from Amazon) and the Ti-89
> > > calculator solved it as well. I think you'll find having one or all of
> > > those on hand will be a tremendous convenience if you run into this
> > > kind
> > > of question often.
> >
> > Wolfram's online integrator is also handy.
> >
> > Interestingly, it can resolve 1/(x*ln(x)) but
> > not 1/ln(x^x).
> >
> >
>
>>http://integrals.wolfram.com/index.jsp?expr=1%2F%28x*ln%28x%29%29&random=fal
>>se
> >
> >
>
> So wolfram is smart, like Maple ... here is Maple:
>
> int(1/x/log(x),x);
> ln(ln(x))
> int(1/log(x^x),x);
> / 1 \
> int|------, x|
> | / x\ |
> \ln\x / /
> int(1/log(x^x),x) assuming x>0;
> ln(ln(x))

Indeed: if you're using principal branches, log(x^x) = x log(x) only when
-pi < Im(x log(x)) <= pi. For example, it's false for real x < -1, where
Im(x log(x)) = pi x. The general formula is log(x^x) = x log(x) + 2 pi i n
where n is chosen so -pi < Im(x log(x)) + 2 pi n <= pi.
And for n <> 0, 1/(x log(x) + 2 pi i n) does not have an elementary
antiderivative.
--
Robert Israel israel(a)math.MyUniversitysInitials.ca
Department of Mathematics http://www.math.ubc.ca/~israel
University of British Columbia Vancouver, BC, Canada
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