Integral pose regression combines an implicit heatmap with end-to-end training for human body and hand pose estimation. Unlike detection-based heatmap methods, which decode final joint positions from the heatmap with a non-differentiable argmax operation, integral regression methods apply a differentiable expectation operation. This paper offers a deep dive into the inference and back-propagation of integral pose regression to better understand the differences in performance and training compared to detection-based methods. For inference, we give theoretical support why expectation should always be better than the argmax operation, \ie integral regression should always outperform detection. Yet in practice, this is observed only in hard cases because the heatmap activation for regression shrinks in easy cases. We then experimentally show that the activation shrinkage is one of the leading causes for integral regression's inferior performance. For back-propagation, we theoretically and empirically analyze the gradients to explain the slow training speed for integral regression. Based on these findings, we incorporate the supervision of spatial prior to speed up training and improve performance.