Here is the preface to the third edition.
This book is intended to serve as a text for the course in analysis that is usually taken by advanced undergraduates or by first-year students who study mathematics.
The present edition covers essentially the same topics as the second one, with some additions, a few minor omissions, and considerable rearrangement. I hope that these changes will make the material more accessible and more attractive to the students who take such a course.
Experience has convinced me that it is pedagogically unsound (though logically correct) to start off with the construction of the real numbers from the rational ones. At the beginning, most students simply fail to appreciate the need for doing this. Accordingly, the real number system is introduced as an ordered field with the least-upper-bound property, and a few interesting applications of this property are quickly made. However, Dedekind's construction is not omitted. It is now in an Appendix to Chapter $1$, where it may be studied and enjoyed whenever the time seems ripe.
The material on functions of several variables is almost completely rewritten, with many details filled in, and with more examples and more motivation. The proof of the inverse function theorem - the key item in Chapter $9$ - is simplified by means of the fixed point theorem about contraction mappings. Differential forms are discussed in much greater detail. Several applications of Stokes' theorem are included.
As regards other changes, the chapter on the Riemann-Stieltjes integral has been trimmed a bit, a short do-it-yourself section on the gamma function has been added to Chapter $8$, and there is a large number of new exercises, most of them with fairly detailed hints.
I have also included several references to articles appearing in the American Mathematical Monthly and in Mathematics Magazine, in hope that students will develop the habit of looking into the journal literature. Most of these references were kindly supplied by R. B. Burckel.
Over the years, many people, students as well as teachers, have sent me corrections, criticisms, and other comments concerning the previous editions of the book. I have appreciated these, and I take this opportunity to express my sincere thanks to all who have written me.
WALTER RUDIN
Best Answer
There is a solitary entry in the index for "indefinite integral" in Walter Rudin's more advanced text (i.e., not the baby one) Real and Complex Analysis.
I quote:
That is the usage I have always followed. In the same spirit I would write $$F(x) = \int_a^x f(t)\,dt$$ for a Lebesgue integrable function $f:[a,b]\to\mathbb R$ and say that $F$ is an indefinite integral for $f$. That allows you to declare that a function $F$ is an indefinite integral in the Lebesgue sense if and only if it is absolutely continuous. Or a function $F$ is an indefinite integral in the Lebesgue sense of a bounded function if and only if it is Lipschitz. (You can even ask for necessary and sufficient conditions for a function to an indefinite integral in the Riemann sense, but that is rather trickier.)
I presume you are thinking instead of a definition that one dimly recalls from those months in a calculus class when hormones and social pressures interfered with comprehension:
Do we need that definition in analysis? Really? It is a good idea, however, to remember it. You may well end up tutoring or TAing or even (horror) lecturing to a large, noisy class of freshman students on the subject of "the calculus."
For the purposes of analysis "indefinite integral" in that sense is not a useful terminology. Just say $f$ is a derivative, $F$ is a primitive for $f$. I don't recall any advanced discussions of derivatives reverting to the terms from the calculus. In Andy Bruckner's large monograph "Differentiation of Real Functions" there is no index but I would wager (a small amount) that the phrase "indefinite integral" does not appear in the calculus sense.
POSTSCRIPT. While I have reproduced the calculus definition for indefinite integral I should comment that the real situation is that calculus students only vaguely understand it anyway. One sees everywhere in online groups the statement: $\int\frac1x\,dx = \ln|x|+ C$. This makes sense on $(0,\infty)$ and on $(-\infty,0)$ according to the definition. It makes no sense on $(-\infty,0)\cup(0,\infty)$ since one arbitrary constant does not suffice and the definition applies only to a single open interval in any case.