In this paper, we use basic asymptotic analysis to establish some uniform asymptotic formulas for the Fourier coefficients of the inverse of Jacobi theta functions. In particular, we answer and improve some problems suggested and investigated by Bringmann, Manschot, and Dousse. As applications, we establish the asymptotic monotonicity properties for the rank and crank of the integer partitions introduced and investigated by Dyson, Andrews, and Garvan.