# Problem with the definition of the Indefinite integral

In the book Thomas' Calculus, the Indefinite integral is defined as follows:

we defined the indefinite integral of the function ƒ with respect to x as the set of all antiderivatives of ƒ, symbolized by $$\int{ƒ(x)dx}$$. Since any two antiderivatives of ƒ differ by a constant, the indefinite integral $$\int$$ notation means that for any antiderivative F of ƒ , $$\int{f(x)dx=F(x)+C}$$

This directly implies that:

$$\frac{dy}{dx}=f(x) \implies y(x)=\int{f(x)dx} = F(x)+C \tag{1}$$

However, in the book Differential Equations With Applications and Historical Notes, Simmons notes:

We solve it by writing: $$y(x)=\int{f(x)dx} + c \tag{2}$$

The 'it' referring to the differential equation in (1).

Question: Isn't the $$c$$ redundant? Since the Indefinite integral already contains the collection of antiderivatives, one would think that would be the case, however, from the first fundamental theorem of calculus we get: $$\int_{a}^{x}{f(t)dt}=\int{f(x)dx} \tag{3}$$

and by substituting (3) in (1), $$y(x)=\int_{a}^{x}{f(t)dt}=F(x)-F(a) \tag{4}$$

And now by comparing (1) and (4) we come to the conclusion that: $$F(x)+C=F(x)-F(a)$$, which clearly has to be False, since a is fixed but C is an arbitrary constant. The only way to make it balanced is to add another arbitrary constant to make: $$F(x)+C=F(x)-F(a) + C_1$$. But in order to make this happen we have to add the $$C_1 to \int{f(x)dx}$$ in (1) and hence confirming the validity of (2). So the $$c$$ in (2) seems necessary again.

Careful, all the second fundamental theorem says is that $$\int_{a}^{x}f(t)\,dt$$ is an antiderivative for $$f(x)$$. That is not the same as saying that $$\int_{a}^{x}{f(t)\,dt}=\int{f(x)\,dx}$$ which is not true. What you can write is $$\int f(x)\,dx = \int_{a}^{x}f(t)\,dt + C.$$ In any case, yes it is generally considered redundant to write $$\int f(x)\,dx + c,$$ but you would do well to double check how Simmons defines $$\int f(x)\,dx.$$ I would be surprised if their definition varies from the standard, so it may be the case that writing the $$+C$$ is just their way of making sure that the reader remembers the need for an arbitrary constant. There is certainly nothing incorrect in writing $$\int f(x)\,dx + c$$ because as you have pointed out, if $$F(x)$$ is an antiderivative of $$f(x)$$ then $$\int f(x)\,dx = F(x) + C,$$ and since the sum of two arbitrary constants is again an arbitrary constant it is still valid to write $$\int f(x)\,dx + c.$$
• Since $\int_{a}^{x}{f(t)dt}$ is a function of $x$, consider $y(x)=\int_{a}^{x}{f(t)dt}$ and use (1). This will prove that $\int_{a}^{x}{f(t)dt}=\int{f(x)dx}$ Aug 24, 2021 at 19:10