Based on the recently developed Finite Integration Method(FIM)for solving one dimensional ordinary and partial differential equations,the present work extends FIM to both stationary and transient heat conduction inverse problems for anisotropic and functionally graded materials with high degree of accuracy.Lagrange series approximation is applied to determine the first order of integral and differential matrices which are used to form the system equation matrix for two and three dimensional problems.Singular Value Decomposition(SVD)is applied to solve the ill-conditioned system of algebraic equations obtained from the integral equation,boundary conditions and scattered temperature measurements.The convergence and accuracy of FIM are investigated with three examples for anisotropic media and functionally graded materials.