The reason why auto-diff is exact and amazing is because (1) the derivative operation is "decomposable" through the chain rule and (2) we've found the symbolic

The reason why auto-diff is exact and amazing is because (1) the derivative operation is "decomposable" through the chain rule and (2) we've found the symbolic derivative of the basic functions. There's not a mathematical trick in auto-diff which we recently discovered. It's mainly a reformulation of the properties of the derivative which turns out to be useful for computers. Finding an "auto-integrate method" would probably involve finding a way of calculating the integral in a decomposable way, and that indeed would be amazing, but I don't really see that happening any time soon.