A generalized function with parameters for both the upper and lower lines of the integrand to use

This is my first Posting. My problem is to find an extreme value of an integral function:

And of course we want to make sure that thisgeneralized function is convergent.Within my ability, the example is given In the attachment.If h itself is small, and the upper and lower limits of the integral are also less affected by h, can we simplify the problem?

Postscript: The problem can be better solved by establishing symbolic variables and then using optimization algorithm. However, the answer method is more concise and efficient under some special solutions, and there is no advantage or disadvantage between the two ideas

example.nb (12.8 KB)

Thank you for your reply. Although I have solved this problem by using matlab’s symbolic variable heuristic a few days ago. But your program has sparked my interest in alternative ideas for this type of problem. Thank you again