Try differentiating both sides and re-arrange to get a DE (not necessarily a nice one) and then use a numerical integration scheme.
I have a definite integral defined by
where is a continuous function of a variable , and and are known numbers. I want to minimize , that is I want to find a continuous function that makes minimum. Ideally I would differentiate it and equate to zero, but because is too complicated to be obtained and then differentiated analytically, I would like to know if there is a numeric technique or any other technique by which this problem can be solved.
Your question falls within the subject "Calculus of Variations", and there is only one thing that I know about it: the Euler–Lagrange equation, which you can read about here:
Hope that helps....
P.S. the link text should be "Euler-Lagrange equation". If you know how to fix it, please let me know.