In the integration of a function, Where does the extra constant of the solution come from?

Constant of integration comes from the fact that derivative of a constant is 0

So if you have two functions:
f(x) = x^2 + 5
g(x) = x^2 - 3

What is f'(x)? What is g'(x)?
f'(x) = 2x, g'(x) = 2x

So now I ask, what is the indefinite integral (anti-derivative) of 2x?
Is it x^2 + 5
Is it x^2 − 3
Or is it some other function of the form x^2 + k, where k is a constant?

It could be any of these, since derivative of x^2 + k is always 2x,
no matter what the value of constant k.
We can't really tell what value of k is unless we are given an initial condition
(such as y(0) = 5 or y(0) = −3), so we leave constant k undefined.


However, when taking definite integrals to calculate area under curve, the constant of integration is not necessary, but using it will not change the result:

For example, if you wanted to find the area under constant function y = c
from x = 0 to x = a, we obviously get a rectangle of width a and heigh c
so area under curve = a*c

Using integration we get:
∫ [0 to a] c dx = cx | [0 to a] = ac − 0c = ac

If we include constant of integration, we get:
∫ [0 to a] c dx = (cx + k) | [0 to a] = (ac + k) − (0c + k) = ac + k − 0 − k = ac


As for the differential equation, units for g is NOT number of atoms
ln N = -λt + g
N = e^(−λt + g) = e^(−λt) * e^g

Since e^g is just an arbitrary constant, we can replace it with another arbitrary constant
Call this new arbitrary constant k. So we get:

N = k e^(−λt), where e^g = k, and k = N₀

f(x)= c1

integral f(x) dx = c1x + c2 <--------------------(you need a new constant of integration)