Do I need different constant names for different levels of integration?

Question

I'm just getting my feet wet in integration, so pardon me if I misuse a term.

Let's take the anti-triple-derivative (I'm not sure if that's what it's actually called) of $8x$.

$y'''=8x$

$y''=4x^2+c$

$y'=\frac43x^3+cx+d$

$y=\frac13x^4+cx^2+dx+e$

Now, could (should) I have used $c$ for all of the constants because they could be anything? For example, $y'=\frac43x^3+cx+c$ and $y=\frac13x^4+cx^2+cx+c$. Or are the different constants necessary because the constants are not necessarily the same?

Answer 1

Different constants are necessary because the constants are not necessarily the same. Well done: $y = \frac 13x^4 + cx^2 + dx + e$ is correct.