QM - Etop of electron distribution of a semiconductor

[f_xer]

Homework Statement

I'm trying to find energy level above Ec where electron distribution makes a peak for a nondegenerate semiconductor. For this case we may take GaAs having Eg = 1.42eV at T = 300K.

Homework Equations

m_e=single isotrophic effective mass or m_0
energy states, g_{c}(E) = \frac{m_{e}\ast\sqrt{2m_{e}(E-E_{c})}}{pi^2 * hbar^3}
fermi function for a nondegenerate semiconductor, f(E) = exp((E_f-E)/kT)
electron distribution, n=N_{c}*exp((Ef-Ec)/kT) and N_{c}=4.21\ast10^{17} cm^-3

The Attempt at a Solution

I think I'll give a fermi energy level equal to 3kT above Ec where semi.con. is still nondegenerate. Then I'll calculate n. Afterwards I'll equate n to \int g_{c}(E)*f(E)*dE taking a limit to 99 % of n. By that I intend to find top limit of the integral which must be the Etop.
But i do not how to evaluate a integral such as \sqrt{E}*exp(c*E)
ps: partial integral is not working.
Is there another {easy :( }approach?

 

[f_xer]

I found the answer:
derivative of
g_{c}(E) * f(E)
gives the minimum points of electron distribution
one of them is E_{c} and the other is E_{top} which is asked by the question ;)