Given a definite integral $I=\int_a^b f(x)\text{d}x$ for some function $f$, we can make any number of substitutions $x=u(x)$ which produce symbolically new integrals, all of which evaluate to the same value.
My main question is this: is it possible to have two definite integrals both of which give the same value yet each of which can not be transformed (i.e. made to look the same symbolically) into the other by substitution?
I get the impression the answer to my question is YES, but I'm not sure. Is there any information or theory out there that deals with this? How can we know when the transformation can not be done (if indeed it cannot be done)?