Help solving this integral equation?

Hi I need help solving the above integral equation. Here f is Fermi Dirac distribution and Nt is Gaussian distribution. n_total, N_0 and N_T are all non values. I need to solve for E_f. Any help is appreciated.

 N_T=@(E) N_t*exp(-E.^2/(2*E_t^2));
 f=@(E) 1/(1+exp((E-Ef)/(k*T)));

Answer 1

[Walter Roberson]

syms E E_F E_t eta_total  k N_0 N_t t
N_T = N_t * exp(-E.^2/(2*E_t^2))

N_T = 

f = 1/(1+exp((E-E_F)/(k*t)))

f = 

eta_mobile = int(N_0 * f, E, -inf, 0)

eta_mobile = 

eta_trap = int(N_t*f, E, 0, inf)

eta_trap = 

eqn = eta_total == eta_mobile + eta_trap

eqn = 

lr = lhs(eqn) - rhs(eqn)

lr = 

specific_lr = subs(lr, [eta_total, N_0, N_t, k, t], [101, 17, 33, 87, 5])

specific_lr = 

fun = matlabFunction(specific_lr, 'vars', E_F)

fun = function_handle with value:
    @(E_F)-integral(@(E)1.7e+1./(exp(E_F.*(-1.0./4.35e+2)+E./4.35e+2)+1.0),-Inf,0.0)-integral(@(E)3.3e+1./(exp(E_F.*(-1.0./4.35e+2)+E./4.35e+2)+1.0),0.0,Inf)+1.01e+2

sol = fsolve(fun, 1)

Warning: Minimum step size reached near x = -Inf. There may be a singularity, or the tolerances may be too tight for this problem.

Warning: Minimum step size reached near x = -Inf. There may be a singularity, or the tolerances may be too tight for this problem.

No solution found.

fsolve stopped because the problem appears regular as measured by the gradient,
but the vector of function values is not near zero as measured by the
value of the function tolerance.

sol = 1

[Arka Bhattacharya]

Thanks for the answer. After exchanging the limits of integral for eta_mobile and eta_trap, I was able to get a solution.