This paper addresses both direct and inverse problems for the bi-Laplace equation in a simply connected, bounded domain. Using Green's formula, the direct problem is recast as a system of Fredholm integral equations. The existence and uniqueness of solution are rigorously established within suitable Sobolev spaces. The objective of the inverse problem is completing missing Cauchy data on a non-measurable boundary portion subjected to Riquier-Neumann boundary conditions. The approach involves reformulating the problem into a set of nonlinear, ill-posed integral equations that faithfully represent the original inverse problem. To address the challenge posed by incomplete Cauchy data, we employ Tikhonov regularization to achieve a robust solution. The proposed methodology is validated through a series of numerical experiments, showcasing its reliability and accuracy in data completion problem.