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Re: Integral Questions

  • To: mathgroup at smc.vnet.net
  • Subject: [mg89343] Re: Integral Questions
  • From: Jean-Marc Gulliet <jeanmarc.gulliet at gmail.com>
  • Date: Sat, 7 Jun 2008 02:50:51 -0400 (EDT)
  • Organization: The Open University, Milton Keynes, UK
  • References: <g2b4oj$nm8$1@smc.vnet.net>

Graeme Dennes wrote:

> I am attempting to calculate the integrals with respect to x of the two
> functions:
> 
>  
> 
>  
> 
> 1.
> 
>          x^2 + a
> 
> arcsin (---------)    -   c 
> 
>            bx
> 
> -----------------------------
> 
>            x
> 
>  
> 
>  
> 
> 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]
> 
>  
> 
> 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?
> 
> 2.  Is there some technique required to cause Mathematica to read the syntax
> 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
> 

Rather than using your own fantasy notation and hopping to find some way 
to coerce Mathematica to understand it, you should pay close attention 
to Mathematica syntax and use it. The following expressions are examples 
of correct Mathematica syntax:

Assuming[{Element[{a, b, c}, Reals], b > 0},
    Integrate[(ArcSin[a/b/x + x/b] - c)/x, x]]

Integrate[(ArcSin[(x^2 + a)/(b*x)] - c)/x, x]

Regards,
-- Jean-Marc


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