Solving an integral equation numerically for an unknown within the integral

Question

I'm trying to solve an integral equation of the form Constant == Integrate [g(x)f(x,Efermi), {x,-200,200}], for the parameter Efermi. The function g(x) is Exp[-(x^2)/5000] and f(x) is 1/(1+Exp[(x-Efermi)/25].

In addition to this, if the Constant is an array, I'd like to extract Efermi as an array. Any inputs on how to go about solving this would be appreciated.

Answer 1

For the scalar case:

Set up a function, h, depending on EFermi, and use FindRoot to solve h[EFermi]==c

Clear[g, f, h];
g[x_] = Exp[-x^2/5000];
f[x_, EFermi_] = (1 + Exp[(x - EFermi)/25])^-1;
h[EFermi_?NumericQ] := NIntegrate[g[x] f[x, EFermi], {x, -200, 200}]
c = 100;
sol = FindRoot[h[e] == c, {e, 10}]
h[e] /. sol
(*  {e -> 55.2154} *)
(*  100. *)
Plot[{h[e], c}, {e, -500, 500}, Epilog -> Point[ {e, c} /. sol],ImageSize -> 300]