Simple math problem. What does this equal? 
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It does in this problem. Sum Integral notation is universal int(f dx) if you interchange dx and f , f will be get out from integral like http://f.int (dx)
the higher you get in math you start to think of integration as an action on a function (linear operator).. Since we always talk about left-actions of operators it makes sense to think of the object ∫dx as "acting" on f(x)
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This is exactly the reason why I *wouldn't* let you do it.. because I think of ∫ acting on differential forms, and it is weird to write the coefficiant of dx after it :P
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you just think of iterated integration as successive action of a linear operator
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Alright, lets define a new integration type with this property named Berger&Mert integral. Problem solved! :)
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Haha but this is what most physicists do... It's nothing new. You just have to learn that notationally, they mean the same as what we do when we stick things inside the integral :) (There are differences of notation everywhere that can nevertheless be self-consistent)
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It's lind of like a wrap around operator.
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