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I am trying to evaluate a numerical integration (to get an expression of the constant Ω)

Suppose that

h[x_] := 1/(1 + a x^2)

and

P=Integrate[x/(h[x] Exp[x/(h[x]) Ω] - 1), {x, 0, ∞}]

where a and Ω are constants. My aim is to get an expression for Ω in terms of a and P after evaluating the integral, but the Mathematica is getting stuck.

Any suggestion to do the above or simplify the problem will be extremely appreciated.

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If I understood correctly, you are trying to solve Ω in terms of a and the result of the integral P.

I'd do something like this.

f[a_?NumericQ, P_?NumericQ] := Module[{},
  h[x_]:= 1/(1 + a x^2);
  FindRoot[
    NIntegrate[x/(h[x] Exp[x/(h[x]) Ω] - 1),{x, 0, Infinity}] == P, 
  {Ω, 2}]] // Quiet   (*Maybe you would like to change the initial guess*)

f[3,0.6]
(*{Ω -> 1.2752}*)
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    $\begingroup$ Thanks a lot Ivan for your suggestion. Yes, I want to get Ω in terms of a and P. So, if I want to get such an expression of Ω (not a value evaluated at specific a and P), I am a bit confused if I can still use NIntegrate. $\endgroup$ – reach2brb Apr 12 '15 at 20:27
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    $\begingroup$ @reach2brb You're welcome! I'm not sure if that integral has a closed form. If it doesn't, you can't have an expression and your only hope is to find the values numerically, like I did. $\endgroup$ – Ivan Apr 12 '15 at 20:41

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