# A simple question about the antiderivative and the integration constant

Let

$$\int f(x) \ dx = F(x) + C$$

There are two possibilities that I am considering.

1) $$F(x)$$ denotes the antiderivative of $$f(x)$$ without any constant term.

2) $$F(x)$$ may include a constant term; the equation above implies that there exist infinitely many solutions, let's say as a form of a function called $$G(x)$$, to the differential equation

$$\frac{d G(x)}{dx}=f(x)$$

in which $$G(x) - F(x)$$ can be expressed by a constant term (e.g. real number) only; the reason why people usually refer to $$F(x)$$ as a function with no constant term is for the sake of convenience and perhaps due to convention.

I am not talking about the effectiveness or meaningfulness of such definitions, i.e. I wonder if 2) is true though this interpretation does not contribute any to solving real maths problems.

I think your option 2 is better, but it is a little confusingly written. The sentence "$$F$$ can include a constant term" is not really meaningful, for a simple reason.

There is no such thing as an "antiderivative without any constant term" or a "function that includes (or does not include) a constant term".

Let me explain.

For a (nice enough) function $$f$$, the set

$$\left\{F(x)\left| \frac{dF}{dx} = f\right.\right\}$$

contains infinitely many functions. The difference of any two of them is a constant function. However, in the set, all functions are "equal", i.e. there isn't "one" among them that has a constant term.

Now, you might think that if $$f(x)=x$$, that the function $$F(x)=\frac{x^2}{2}$$ is the "antiderivative without the constant term", but you'd be wrong. It's certainly the antiderivative that is easiest to write down, but that's just because we have a simple way of writing simple polynomials.

For example, think about the functions $$\cosh$$ and $$\sinh$$. For these two functions, we know that $$\cosh^2(x)-\sinh^2(x) = 1$$, which means that, for any constant $$C_1$$, there exists some constant $$C_2$$ such that

$$\cosh^2(x) + C_1 = \sinh^2(x) + C_2$$

which means that, for the function $$f(x)=\frac{d}{dx}(\sinh^2(x)) =\frac{d}{dx}(\cosh^2(x))$$, there isn't really any function that is the "antiderivative without the constant term".

As a consequence, there is also no such thing as the concept of one function "including a constant term".

What the expression $$\int f(x) dx = F(x) + C$$ means is that $$F$$ is one possible function which has the derivative $$f$$, and that all other derivatives differ from $$F$$ by a constant.

• Thank you. Could you please elaborating more on the example with $\cosh(x)$ and $\sinh(x)$? I think I didn't understand that properly - the "antiderivative without the constant term". – curious Jan 21 '20 at 12:37
• @curious If $f(x)=\frac{d}{dx}(\sinh^2(x))$, then $f(x) =\frac{d}{dx}(\cosh^2(x))$. In this case, what is $\int f(x)dx$? Is it $\sinh^2(x)+C$, or is it $\cosh^2(x)+C$? – 5xum Jan 21 '20 at 12:43
• Clearly $$\frac{1}{2}(\cosh^2 x + \sinh^2 x)$$ is the one without constant term ;) – Daniel Fischer Jan 21 '20 at 13:23