Why is it important to introduce constant of integration immediately when the integration is performed?

I'm assuming you are asking why you cannot work through your simplification and only put a constant on the last line.

The simplest answer is that mathematicians are picky people, and when working through a problem EVERY line must make absolute mathematical sense. Leaving the constant off until the last line makes every line between the point where the integration occurs and the last line false. (Unless you are lucky and the constant of integration is 0, however this still needs to be proven)