In this paper, we address the problem of finding the optimal trajectory for an unmanned aerial vehicle (UAV) for improving the wireless coverage of a terrestrial cellular network. In particular, we consider a UAV that is tasked to travel from one point to another within a given time constraint, and it can also simultaneously assist the cellular network by providing wireless coverage during its mission. Considering an interference limited downlink of a cellular network, we formulate an optimization problem for maximizing the proportional-fair (PF) data rate of the cellular network and explore dynamic programming (DP) technique for finding the optimum UAV trajectory. We also explore the optimal UAV trajectories associated with maximum sum-rate and 5th percentile spectral efficiency (5pSE) rate and compare the capacity and coverage performance of the three approaches. Numerical simulations show that the maximum sum-rate trajectory provides the best per user capacity whereas, the optimal PF trajectory yields higher coverage probability than the other two trajectories. The optimal trajectories are generally infeasible to follow exactly as the UAVs can not take sharp turns due to kinematic constraints. Hence, we generate smooth trajectories using Bezier curve.