Blood flow-through segments of large arteries of man, between adjacent bifurcations, can be modeled as pulsatile flow in tapered converging tubes, of small angle of convergence, up to 2 deg. Assuming linearity, rigid tube and homogeneous Newtonian fluid, the physiological flow field is governed by the Navier-Stokes equation with dominant nonlinear and unsteady terms. Analytical solution of this problem is presented based on an integral method technique. The solution shows that even for small tapering the flow pattern is markedly different from the flow obtained for a uniform tube. The periodic shear stresses at the wall and pressure gradients increase both in their mean value and amplitude with increased distance downstream. These results are highly significant in the process of atherogenesis.