Purpose Of Adding A Constant After Integrating A Function I would like to know the whole purpose of adding a constant termed constant of integration everytime we integrate an indefinite integral $\int f(x)dx$. I am aware that this constant "goes away" when evaluating definite integral $\int_{a}^{b}f(x)dx $. What has that constant have to do with anything? Why is it termed as the constant of integration? Where does it come from?
The motivation for asking this question actually comes from solving a differential equation $$x \frac{dy}{dx} = 5x^3 + 4$$ By separation of $dy$ and $dx$ and integrating both sides, $$\int dy = \int\left(5x^2 + \frac{4}{x}\right)dx$$  yields $$y = \frac{5x^3}{3} + 4 \ln(x) + C .$$
I've understood that $\int dy$ represents adding infinitesimal quantity of $dy$'s yielding $y$ but I'am doubtful about the arbitrary constant $C$.

Say every month you earned 10 dollars on a principal of 1000 dollars and put it aside. What have you got after 10 months ?
Not always 100 dollars or 180 dollars or 130. That is so, only if the box  contained nothing not at start.
If it contained 35 dollars at start, that is the constant added to the accumulated (integrated) amount $A:$
$ A =  m.10 + c =  m.10 + 35 $ dollars or 135 or 235 dollars..
The same applies to more complicated integrals. It is also the initial value in integration of a variable function with initial  value prescribed to evaluate boundary value differential equations.
