Is the notion of indefinite integrals really necessary in analysis? (Walter Rudin "Principles of Mathematical Analysis 3rd Edition") I am reading "Principles of Mathematical Analysis 3rd Edition" by Walter Rudin.
The author didn't write the notion of indefinite integrals in this book at all.
This book is very famous and has influence in analysis.
Is the notion of indefinite integrals really necessary in analysis?
 A: 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:

"If $f\in L^1(\mathbb R^k)$ and $$\mu(E) = \int_E f(x)\,dx$$ it is
reasonable to call $\mu$ the indefinite integral of $f$."

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:

Definition.   Suppose that $f:(a,b)\to \mathbb R$ has the property that there exists a function $F:(a,b)\to\mathbb R$ such that
$F'(x)=f(x)$ for all $a<x<b$.  Then $F$ is called a primitive [or
antiderivative] for $f$ on that open interval and the expression
$$\int f(x)\,dx =F(x) + C$$ is called an indefinite integral with
the understanding that $C$ is an arbitrary constant.

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.
