  Analytical solutions are presented for eigenvalues, eigenfunctions of
{\color{red} D-dimensional Schrodinger equation having Eckart potential} within
Nikiforov-Uvarov method. This uses a new, improved approximation for
centrifugal term, from a combination of Greene-Aldrich and Pekeris
approximations. Solutions are obtained in terms of hypergeometric functions. It
facilitates an accurate representation in entire domain. Its validity is
illustrated for energies in an arbitrary $\ell \neq 0$ quantum state. Results
are compared for a chosen set of potential parameters in different dimensions.
In short, a simple accurate approximation is offered for Eckart and other
potentials in quantum mechanics, in higher dimension.
