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I need to solve the following equation, for which I am using Solve as below

λ = 20;
μ =  3.1623;

Solve[
      Integrate[Exp[-λ π km r^2 ArcTan[r Sqrt[μ]]] 2 Gamma[λ π + 1] / Gamma[λ π]
                r (1 - r^2)^(-1 + λ π), {r, 0, Infinity}] == 0.5,
      km]

But Mathematica does not output anything. Any help?

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Solve and Integrate try to solve for the exact value, which takes a lot of time or, in some cases, is impossible. You may want to use NSolve and NIntegrate.

λ = 20;
μ = 3.1623;
f[km_Real] := NIntegrate[Exp[-λ*π*km*r^2*ArcTan[r*Sqrt[μ]]]
 *2*Gamma[λ*π + 1]/Gamma[λ*π]*r*(1 - r^2)^(-1 + λ*π), {r, 0, Infinity}];
NSolve[f[km]==.5, km]
(* {{km -> 5.25146}} *)
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  • $\begingroup$ Thanks for your answer. It works for me! However, if I want to find the expression for $km$, it is possible? $\endgroup$ – Srestha Narayanan Apr 22 '16 at 2:33
  • $\begingroup$ It seems impossible. Mathematica is unable to calculate the integral Integrate[Exp[-λ*π*km*r^2*ArcTan[r*Sqrt[μ]]]*2*Gamma[λ*π + 1]/Gamma[λ*π]*r*(1 - r^2)^(-1 + λ*π), {r, 0, Infinity}]. Inputting it outputs the integral itself, not a simplified equation. That is, Mathematica cannot create an inverse function and give the exact expression for km. $\endgroup$ – JungHwan Min Apr 22 '16 at 2:49

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