I sometimes find writing and keeping track of the constants of integration a somewhat messy job. Yes, sometimes it's necessary but in many situations that I come across in my level of mathematics, it is a waste of time and space.
An exagerated example: $$f''(x) = g''(x)\\ \stackrel{\int\text{ing}}{\implies} f'(x) + C_1 = g'(x) + C_2\\ \stackrel{\int\text{ing}}{\implies} f(x) + C_3 + C_1x + C_4= g(x) + C_5 + C_2x + C_6\\ \implies f(x) = g(x) + C_ax + C_b $$ where $C_a = C_2 - C_1$ and $C_b = C_6 + C_5 - C_4 - C_3$
This just seems like a ridiculous amount of tracking in certain cases and I often just combine as many constants in one shot without mercy in most problems I face and never explain the sources of the constants (because it doesn't really matter)
$\times$ Some people I know avoid distinguishing the constants by just marking them all as $C$ and only distinguish between coefficients and constants as I've illustrated in the last step of my example.
$\times$ Some other students leave out the constants all together and write a " + C " only in last step but they don't realize that they are often neglecting terms where the constants turn into coefficients.
$\Large\star$ It isn't a mystery as to why the constant tickles people's lazy bones. The lethargic attitude almost everyone shows toward it is because that " + C " is just annoying. Sure, it may seem necessary but what about in simplifications?
$$ \begin{align} \int f'(x)\ \mathrm dx &= f(x) + C \\ &= f_1(x) + C \\ &= f_2 (x) + C \\ &= f_3 (x) + C \\ &= \dots \\ &= f_n(x) + C\\ \end{align}$$
where $f_{k\in \mathbb N}(x)$ is a simplified form of $f(x)$
It's usage is monotonous and seems absolutely unnecessary during simplification.
My question is hence this:
How can one safely hide the constant of integration during simplification of equations and what are the restrictions in such hiding?
Thank you in advance.
Edit: I am seeking a notation which would validly allow for a neglection of the constant during simplification of differential equations. Imagine 100 steps of simplification. The +C would be annoying. Apologies if this point was not clear through my rant.
...I often just combine as many constants in one shot without mercy...
Yes, that's what I do. But that one constant bothers me during simplification. $\endgroup$