A novel linear tracking integrator (LTI) with integral compensation is proposed for efficient integral estimation from a contaminated measurement with a constant or time-varying bias. The limitation of finite-time convergent integral observer (FTCIO) in ruling out the integral drift is firstly revealed via describing function method. Subsequently, by the utilization of integral action in the feedback path, a simple but effective linear tracking integrator is established to provide a practical solution in achieving a drift-free integral estimate. The highlight is that the proposed LTI can simultaneously give the accurate integral and tracking estimates from a noisy measurement without relying on the condition of observability. In addition, frequency-domain analysis of LTI is investigated to give a viable guideline of parameter tuning. Illustrative simulations and comparison with Kalman filter are included to demonstrate the superiority of LTI in accomplishing precise integral tracking in the presence of constant or time-varying bias. Finally, the effectiveness of LTI is also confirmed by an application on autopilot design for aircraft.