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MathGroup Archive 1997

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Re: Unsolved integral

  • To: mathgroup at smc.vnet.net
  • Subject: [mg8879] Re: [mg8826] Unsolved integral
  • From: seanross at worldnet.att.net
  • Date: Tue, 30 Sep 1997 20:16:45 -0400
  • Sender: owner-wri-mathgroup at wolfram.com

guest at uv.es wrote:
> 
> Dear colleagues,
> I'm trying to solve the following integral :
> Integrate[(1-a-x)^(0.5) (1/x)^(0.5)   1/((1-a-x)^2-4*a*b*x)^(0.5)),{x,0,1-a}]
> but Mathematica 3.0 gives it back to me. I would be very appreciate if
> somebody tells me how to solve it.
> Best regards:
> Jose M. Ramon <jose.m.ramon at uv.es>

1)  First, get rid of the clumsy notation and clean up the fractions. 
The more complicated structure you give mathematica, the harder it is to
get a clean symbolic result.  Especially get rid of decimal exponents
when rational exponents are available.  (ie.  x^0.5 is not recommended.
x^(1/2) or Sqrt[x] is much better.)

Integrate[Sqrt[(1-a-x)/( x *((1-a-x)^2-4*a*b*x))],{x,0,1-a}].

2)  Don't put convoluted polynomial expressions into Integrate.  In this
case, 
Sqrt[ExpandDenominator[(1-a-x)/( x *((1-a-x)^2-4*a*b*x))] ]
does the trick. Even in this form, Mathematica won't return a definite
integral.

3) Next, if you can't get a definite integral, try an indefinite one.


Integrate[Sqrt[(1-a-x)/((x - 2 a x + a^2 x - 2 x^2 + 2 a x^2 - 4 a b x^2
+ x^3\)),x]

The result is nearly two pages long and is in terms of polynomials and
Elliptic Integrals of the second and third kind(EllipticF and
EllipticPi).  These Elliptic Integrals are not really solving the
problem since they are integrals in their own right, but they are built
in to mathematica and are fast executing and reasonably robust
numerically.  You can then evaluate the huge expression at zero and 1-a
to get your answer.


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