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Re: Integral Questions

  • To: mathgroup at smc.vnet.net
  • Subject: [mg89351] Re: Integral Questions
  • From: Szabolcs <szhorvat at gmail.com>
  • Date: Sat, 7 Jun 2008 02:56:08 -0400 (EDT)
  • References: <g2b4oj$nm8$1@smc.vnet.net>

On Jun 6, 1:49 pm, "Graeme Dennes" <gden... at bigpond.com> wrote:
> I am attempting to calculate the integrals with respect to x of the two
> functions:
>
> 1.
>
>          x^2 + a
>
> arcsin (---------)    -   c
>
>            bx
>
> -----------------------------
>
>            x

Please use the InputForm of expressions to make copying and pasting
easier.  (Right click -> Copy As -> plain text or input text)

>
> 2. As above, squared.
>
> where a, b, c are constants.
>
> Mathematica 6 presents solutions which seem to depend on the form of the
> function entered, and in some instances, Mathematica does not read the
> syntax correctly.
>
> Eg, using ((x^2 + a)/(bx)) and (x/b + a/b/x) as different (but correct)
> forms of the arcsin function yield different results!
>
> Taking the first function, and entering it as:
>
> Integral[((arcsin(a/b/x+x/b))-c)/x,x]

This syntax is completely incorrect, and it is no wonder Mathematica
did not understand it.  Please go through some of the basic  tutorials
in the documentation.  The correct syntax is

Integrate[(ArcSin[a/b/x + x/b] - c)/x, x]

Mathematica will return this unevaluated because it cannot calculate
this integral.

>
> The answer presented is:   - a arcsin        arcsin x
>
>                              --------   +   --------    -    c Log (x)
>
>                                 bx              b
>
> Note the missing argument of the first arcsin.
>
> As noted, different results can be provided for different (but correct!)
> forms of the entered function. I do not understand why this would be so.
>
> My questions:
>
> 1.  Are there (true) closed form solutions to both functions 1 and 2?

I don't know.  I tried two other respected systems, and neither could
do the integral.

>
> 2.  Is there some technique required to cause Mathematica to read the syntax
> correctly?

Yes, simple: you have to type it correctly.

>
> 3.  Is there some technique required to cause Mathematica to provide the
> correct solution, assuming a closed form solution exists?
>
> 4.  HELP!!
>
> If the answer to Q1 is NO, then that would explain why the correct answer is
> not obtainable.
>
> Any advice with this issue would be much appreciated.
>
> Kindest regards,
>
> Graeme



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