We develop an integral approach to obtain interior a priori \(C^{2}\) estimates for convex solutions of prescribing scalar curvature equations \(\sigma _2(\kappa ) = f(x)\) as well as the Hessian equations \(\sigma _2(D^2u)=f(x)\) . The advantage of this new approach is that only the Lipschitz modules of f(x) is needed for the estimate. As a result, we prove that the \(C^{2}\) modules of the solutions depend only on \(\Vert u\Vert _{C^{1}}, \Vert f^{-1}\Vert _{L^{\infty }}\) , and \(\Vert f\Vert _{C^1}\) instead of \(\Vert f\Vert _{C^k}\) for some \(k\ge 2\) in all the papers we have known up to now.
