Numerical integration of measured signals is challenging due to sensor noise, where sensor bias leads to a spurious ramp, and white noise leads to random-walk divergence. This paper presents a novel approach to numerical integration of sensor data based on adaptive input estimation. In particular, retrospective cost input estimation (RCIE) is applied to a one-step-delayed differentiator model to estimate the unknown input, which is the desired integral of the output. Numerical examples show that, for harmonic signals corrupted by white noise, RCIE integration eliminates the random walk that arises from standard numerical integration.