How paracrine signals are interpreted to yield multiple cell fate decisions in a dynamic context during human development in vivo and in vitro remains poorly understood. Here we report an automated tracking method to follow signaling histories linked to cell fate in large numbers of human pluripotent stem cells (hPSCs). Using an unbiased statistical approach, we discovered that measured BMP signaling history correlates strongly with fate in individual cells. We found that BMP response in hPSCs varies more strongly in the duration of signaling than the level. However, we discovered that both the level and duration of signaling activity control cell fate choices only by changing the time integral of signaling and that duration and level are therefore interchangeable in this context. In a stem cell model for patterning of the human embryo, we showed that signaling histories predict the fate pattern and that the integral model correctly predicts changes in cell fate domains when signaling is perturbed. Using an RNA-seq screen we then found that mechanistically, BMP signaling is integrated by SOX2.