# Solve does not seem to work for this equation

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}} *)

Thanks for your answer. It works for me! However, if I want to find the expression for kmkmkm, it is possible?
– Srestha Narayanan
Apr 22 at 2:33

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.
– JHM
Apr 22 at 2:49