Apologies for a question that I should have been able to answer ages ago, but am having some trouble with!

As part of my day job as a theoretical physics grad student, I've got an integral which, when I factor out various physics-y constants, looks like

[; \int \frac{dx}{1 e^x } ;]

So I can come up with a substitution which solves this, and all is well. The weird thing is that when I do it in a very particular way (which is of course how I did it), an arbitrary constant I put into that substitution doesn't cancel out (it does cancel if I put in limits of integration, but doesn't cancel in the improper integral). Here are the steps as I've done them (a is constant):

[; u = a e^x ;], [; du = u dx ;]

[; \int \frac{dx}{1 e^x} = \int \frac{du}{u}\frac{1}{1 \frac{u}{a}} ;]

[; = \int du \(\frac{1}{u} - \frac{1/a}{1 u/a}\) ;]

[; p = 1 u/a ;], [; dp = du/a ;]

[; \int \frac{dx}{1 e^x} = \int \frac{du}{u} - \int \frac{dp}{p} ;]

[; = \ln{u} - \ln{p} ;]

[; = \ln{\frac{a e^x}{1 e^x}} ;]

I can do the integral in seemingly-equivalent ways that don't lead to this a-dependence, and it's not too worrying since it just amounts to adding a constant of integration anyway, but it's bizarre that an arbitrary factor in a substitution isn't cancelling out. I think my algebra is right. So what's going on?

EDIT: Just so it's clear what I'm asking, I know that the integral I get is correct, but what I find strange is that I'm basically getting a constant of integration out of my choice of a substitution (i.e., my choice of which value of a I pick). I've never seen a constant of integration show up like that before; surely anything that I define as part of my substitution shouldn't show up in the final answer, no? So I'm wondering what's happening here, mathematically.