# 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

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