Or what happens to f(t) when this integral is evaluated? Does it just stay the same? That would be nice if it did, but I imagine something complicated happens to it.
Think of it in terms of something simpler. You know that . But is NOT equal to for every function f! That would make integration trivial. What you are doing is essentially a substitution. And if you let u= f(x), then you also have to let du= f'(x)dx.