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Re: Re: simplify a trig expression
*To*: mathgroup at smc.vnet.net
*Subject*: [mg65449] Re: [mg65436] Re: [mg65415] simplify a trig expression
*From*: Murray Eisenberg <murray at math.umass.edu>
*Date*: Sun, 2 Apr 2006 05:00:09 -0400 (EDT)
*Organization*: Mathematics & Statistics, Univ. of Mass./Amherst
*References*: <200603311109.GAA15029@smc.vnet.net> <200604011038.FAA07301@smc.vnet.net>
*Reply-to*: murray at math.umass.edu
*Sender*: owner-wri-mathgroup at wolfram.com
OK, now how about the following--another case where Mathematica gives a
more complicated looking answer than the typical paper-and-pencil direct
substitution would provide?
Mathematica gives:
Integrate[Sin[x]/(1 - Cos[x]), x]
2*Log[Sin[x/2]]
I'd like an answer in the form of Log[1-Cos[x]] (plus a constant, to
actually equal the above). The best I've been able to do so far is:
(Integrate[Sin[x]/(1 - Cos[x]), x]
// Simplify[# /. a_*Log[b_]:>Log[b^a]] &
//MapAt[TrigReduce, #, 1]&) /. (Log[c_ b_]->Log[b]+Log[c])
-Log[2] + Log[1 - Cos[x]]
Is there some easier way?
Andrzej Kozlowski wrote:
> This is one of those cases where FullSimplify will not work because
> it lacks a suitable transformation function. In this particular case
> the transformation function is of the form:
>
> f[n_*Log[a_]] := Log[a^n]
>
> Of course this is only valid with various assumptions on n and a, but
> I won't bother with this here. Anyway, observe that:
>
>
> FullSimplify[Integrate[Cos[x]/(Sin[x] + 1), x],
> TransformationFunctions -> {Automatic, f}]
>
>
> Log[Sin[x] + 1]
>
> Note also that Simplify will not work even when you add f.
>
> I am not sure if there are good reasons for adding a version of f
> (taking account of suitable assumptions) to the default
> transformation functions of FullSimplify. It may however be a good
> idea to have another possible value for the option
> TransformationFunctions besides only Automatic and user defined ones.
> In fact I have suggested in the past one or two other useful
> TransformationFunctions; perhaps it might be a good idea to define
> more and collect them into a single option value or maybe several.
>
> Andrzej Kozlowski
>
>
>
>
>
>
> On 31 Mar 2006, at 13:09, Murray Eisenberg wrote:
>
>> A direct substitution (with paper and pencil) gives that the
>> integral of
>> Cos[x]/(Sin[x] + 1) is Log[Sin[x] + 1]. This is valid provided
>> Sin[x]
>> is not -1.
>>
>> Mathematica gives:
>>
>> Integrate[Cos[x]/(Sin[x] + 1), x]
>> 2 Log[Cos[x/2] + Sin[x/2]]
>>
>> Is there some simple way to coerce the latter Mathematica-supplied
>> result into the paper-and-pencil answer?
>>
>> The closest I could get is:
>>
>> Log[TrigExpand[Expand[(Cos[x/2] + Sin[x/2])^2]]] /.
>> {Sin[x/2] -> Sqrt[(1 - Cos[x])/2],
>> Cos[x/2] -> Sqrt[(1 + Cos[x])/2]}
>> Log[1 + Sqrt[1 - Cos[x]]*Sqrt[1 + Cos[x]]]
>>
>> Am I not seeing some easier TrigExpand or TrigReduce method?
>>
>> --
>> Murray Eisenberg murray at math.umass.edu
>> Mathematics & Statistics Dept.
>> Lederle Graduate Research Tower phone 413 549-1020 (H)
>> University of Massachusetts 413 545-2859 (W)
>> 710 North Pleasant Street fax 413 545-1801
>> Amherst, MA 01003-9305
>>
>
>
--
Murray Eisenberg murray at math.umass.edu
Mathematics & Statistics Dept.
Lederle Graduate Research Tower phone 413 549-1020 (H)
University of Massachusetts 413 545-2859 (W)
710 North Pleasant Street fax 413 545-1801
Amherst, MA 01003-9305
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