# Textbooks that use notation with explicit argument variable in the upper bound $\int^x$ for “indefinite integrals.”

I dare to ask a question similar to a closed one but more precise.

Are there any established textbooks or other serious published work that use $$\int^x$$ notation instead of $$\int$$ for the so-called "indefinite integrals"?

(I believe I've seen it already somewhere, probably in the Internet, but I cannot find it now.)

So, I am looking for texts where the indefinite integral of $$\cos$$ would be written something like: $$\int^x\cos(t)dt =\sin(x) - C$$ or $$\int^x\cos(x)dx =\sin(x) + C.$$

(This notation looks more sensible and consistent with the one for definite integrals than the common one with bare $$\int$$.)

Some context.

IMO, the indefinite integral of $$f$$ on a given interval $$I$$ of definition of $$f$$ should not be defined as the set of antiderivatives of $$f$$ on $$I$$ but as the set of all functions $$F$$ of the form $$F(x) =\int_a^x f(t)dt + C,\qquad x\in I,$$ with $$a\in I$$ and $$C$$ a constant (or as a certain indefinite particular function of such form). In other words, I think that indefinite integrals should be defined in terms of definite integrals and not in terms of antiderivatives. (After all, the integral sign historically stood for a sum.)

In this case, the fact that the indefinite integral of a continuous function $$f$$ on an interval $$I$$ coincides with the set of antiderivatives of $$f$$ on $$I$$ is the contents of the first and the second fundamental theorems of calculus:

1. the first fundamental theorem of calculus says that every representative of the indefinite integral of $$f$$ on $$I$$ is an antiderivative of $$f$$ on $$I$$, and

2. the second fundamental theorem of calculus says that every antiderivative of $$f$$ on $$I$$ is a representative of the indefinite integral of $$f$$ on $$I$$ (it is an easy corollary of the first one together with the mean value theorem).

• The claim that this looks "much less absurd than the common one with the bare $\int$" is extremely subjective. Addionally, why aren't you satisfied with the answer given in your linked thread? – mrtaurho Jul 20 at 20:32
• @mrtaurho, if we agree on any reasonable measure for evaluating absurdity of mathematical notation, i may be willing to argue with you. – Alexey Jul 20 at 20:37
• @mrtaurho, if you mean "I have a feeling I've seen your notation used elsewhere (the nagging thought at the back of my head is that Russian authors used it, but I haven't got any examples to hand to verify that)", then no, I am not satisfied with this. I do not see any other references in the linked thread. – Alexey Jul 20 at 20:47
• Let me reformulate what I wanted to empahsize: is this controversial (yes, I would call it controversial as you arguing against a really well-established notation) side note really needed? To discuss mathematical notation is, let's say strange; someone invented it and then we either kept it or changed it over time, but afterall there are some - the bare $\int$ among them - established notations. Please, don't get me wrong, I do not dislike your question or want to question your premise. I only find your wording a little bit to extreme. – mrtaurho Jul 20 at 20:59
• @mrtaurho, i changed the wording slightly because i realised that the notations with $\int^x$ or $\int_a^b$ (for definite integrals) are not free of issues either. – Alexey Jul 21 at 7:03

## 1 Answer

That notation is used in the classic textbook Elementary Differential Equations by William E. Boyce and Richard C. DiPrima, at least in the third edition (1976), which is the one I have. Quoting from p. 11 (beginning of Chapter 2):

The simplest type of first order differential equation occurs when $$f$$ depends
only on $$x$$. In this case $$y'=f(x)\tag2$$ and we seek a function $$y=\phi(x)$$ whose derivative is the given function $$f$$. From
elementary calculus we know that $$\phi$$ is an antiderivative of $$f$$, and we write $$y=\phi(x)=\int^xf(t)dt+c,\tag3$$ where $$c$$ is an arbitrary constant. For example, if $$y'=\sin2x,$$ then $$y=\phi(x)=-\frac12\cos2x+c.$$      In Eq. $$(3)$$ and elsewhere in this book we use the notation $$\int^xf(t)dt$$ to denote
an antiderivative of the function $$f$$; that is, $$F(x)=\int^xf(t)dt$$ designates some
particular representative of the class of functions whose derivatives are equal to $$f$$.
All members of this class are included in the expression $$F(x)+c$$, where $$c$$ is an
arbitrary constant.

P.S. On second thought, I'm not sure Boyce & DiPrima use the notation $$\int^xf(t)dt$$ in quite the same way you do. For them the general solution of the differential equation $$y'=f(x)$$ is $$y=\int^xf(t)dt+c$$ since $$y=\int^xf(t)dt$$ is some (unspecified) particular solution; but for you I think $$y=\int^xf(t)dt$$ is already the general solution.

• Thanks, the subtly different meaning is a secondary question. In fact, i cannot quite decide what the indefinite integral (in whatever notation) should mean: a class of functions or an unknown function. I am more favourable to treat is as an unknown function similarly to $O(...)/o(...)$ (big "O" and small "o"). – Alexey Jul 21 at 7:14
• Another remark: they seem to define the indefinite integral in terms of antiderivatives, and indeed this is how it is usually done. I would have preferred it to be defined in terms of definite integrals. – Alexey Jul 21 at 7:28
• The phrase "From elementary calculus we know that ϕ is an antiderivative of f" is strange, however. This is what an antiderivative is by definition, it is enough to know what "antiderivative" means. – Alexey Jul 21 at 7:30