In this paper, we propose a user-friendly integral inequality to study the coupled parabolic chemotaxis system with singular sensitivity under the Neumann boundary condition. Under a low diffusion rate, the classical solution of this system is uniformly bounded. Our proof replies on the construction of the energy functional containing ${\int }_{\mathsf{\Omega }}\frac{{\left|v\right|}^4}{v^2}$ with $v>0$. It is noteworthy that the inequality used in the paper may be applied to study other chemotaxis systems.