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Where does an integration constant go when you integrate inside an exponent

So I've encountered this case a few times before. One clear example is an integrating factor of an first order DE, why is there no integration constant? I thought, since an integral of constant gives 0, the definite integral of integrating factor must include the arbitrary constant term? Could anyone please explain it with an example?
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Pangol
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Original post by delalu
So I've encountered this case a few times before. One clear example is an integrating factor of an first order DE, why is there no integration constant? I thought, since an integral of constant gives 0, the definite integral of integrating factor must include the arbitrary constant term? Could anyone please explain it with an example?


Suppose the integrating factor is (integral of) ef(x) dx, and suppose that the integral of f(x) is F(x), so that the integrating factor is eF(x) + c. This is the same as eF(x)ec, and since ec is a constant, we could call it A and have an integrating factor of AeF(x). If we now multiply the differential equation through by this, we could then divide through by A and it will be as if it was never there. So it's easiest just to not bother with it in the first place.

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