Comment #3 on issue 1360 by asmeurer: "integrate" doesn't allow the use of functions as the variable of integration.
http://code.google.com/p/sympy/issues/detail?id=1360
This can be well defined. See http://en.wikipedia.org/wiki/Riemann-Stieltjes_Integral. The function of integration has to be a bounded variation for it to work.
However, I didn't know that the Reimann-Stieltjes integral had any applications to actual computed integrals. It is a nice theoretical tool, because by choosing different "functions of integration," you can get different integrals. Indeed, you can show that all linear functionals on C[a, b] are just Riemann-Stieltjes integrals. It also provides a way to use integration theory to study summations, since if the function is a step function, the integral reduces to a sum over the jumps.
Actually, I don't know if this can be, or if it has ever been, generalized to a definite integral. Even if it has, I wouldn't know how to compute it. Integral(f(x), g(x)) is the same as Integral(f(x)*g(x).diff(x), x) if g(x).diff(x) is continuous. (See the Wikipedia article).
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