[UPSC CSE 2012] Statistical Physics (Q295, 15M)

Show that both Fermi-Dirac and Bose-Einstein distributions reduce under certain condition in a form which gives the total number of particles as N = A\int_0^\infty \sqrt{\varepsilon}e^{-\beta\varepsilon}\,d\varepsilon where A is a constant and \beta = 1/k_{\mathrm B}T. Further show that this expression is just the same as that obtained from the Maxwellian speed distribution.