How do you show that thermal broadening at Fermi distribution is~3.5kT when the temperature is nonzero and finite?

I searched and found out that the thermal broadening is given as $3bkT/(4a^2)$ , but this way need some data to find out $a$ and $b$ , as the $a$ and $b$ is just a number given by a model. I can't find a clue to find the thermal broadening just with Fermi distribution function and without any experimental data. I am working on this problem by trying to integrate the Fermi distribution function, as it seems to be the only thing I can do with this function, but that's all I could do.