You are assuming that the 2 "c"s are the same. Yes, , for some constant, c. Yes, where "c" and "C" are generally NOT the same. Nowthatreduces to which tells us that is aconstant- NOT "ridiculous".

I don't know what you mean by"purely anti-derivative". It is, of course, an anti-derivateve. What does the "purely" indicate? And integration by substitution makes use of both differentials and derivatives which have a "one-to-one correpondence" so it really doesn't matter.2. Integration by substitution makes use of differentials (not simply derivatives), thus is it purely anti-derivative?

Thank you in advance