Why don't we add multiple integration constants

Question

suppose we had $\int x^2 + x \, dx$, I understand you can split it up into $\int x^2 \, dx$ + $\int x \, dx$, but then why is it that we don't add seperate constants of integration for each indefinite integral, and simply add a single constant on the end once we are done integrating?

Thanks

Answer 1

If indefinite integration is viewed loosely as the process of finding a function given its derivative, then the constant of integration is used to express the fact that there are many functions that have the same derivative (because the derivative of any constant is $0$).

In your example, notice that any function $f$ of the form $$ f(x)=\frac{x^3}{3}+\frac{x^2}{2}+C $$

for some constant $C$ will yield the derivative $f'(x)=x^2+x$. You can define $f$ to be $$ f(x)=\frac{x^3}{3}+\frac{x^2}{2}+A+B+C, $$ for some constants $A,B,C$ and the derivative of $f$ will stay the same.

Answer 2

Writing $\int f(x) dx = F(x) + C$ is short-hand for saying that the family of antiderivatives of $f(x)$ is the set of functions $F(x) + C$, where $C$ is some arbitrary constant.

We could write $\int f(x) dx = F(x) + C_1 + C_2$, but then we'd note that (if there are no restrictions on the constants in question) the two sets are identical - for any function in the latter set defined by specific values of $C_1$ and $C_2$, we can always show that it maps to a value of $C$ and a corresponding function in the former.

In the end, we choose the representation that's most useful - having two constants doesn't add any information in this case, and so we would usually choose to absorb them into a single one. We might choose to leave them separated if having two constants gave additional insight into something - e.g. if we forced $C_1$ to be an integer and $C_2$ to be in the interval $[0, 1)$ and they had distinct impacts on the system we're looking at.

Similarly, the family of strictly positive functions that satisfies $f' = f$ could be written as $f(x) = e^{x+A}$ or $f(x) = Be^x$ (with $B > 0$), and while they represent the same functions the placement of the constant can have a different interpretation.