TITLE: Why doesn't Spivak ever write $dx$ in an integral?
QUESTION [17 upvotes]: I've noticed that Spivak, and many other analysis books I read like Munkres, do not use $dx$ when they integrate. Why is that? This is a serious question.

REPLY [2 votes]: Part of the issue here is that there are really two things going on in basic integral calculus:

A theory of "definite integrals of functions": given a continuous real valued function $f$ on an interval $[a,b]$ we want to associate a real number, the definite integral $\int_a^b f$.
A theory of how to compute these definite integrals using symbolic manipulation (that's what the word "calculus" means). In this setting, we have an expression, which is not the same as a function, and we want to work out a definite integral algorithmically in terms of this expression. This is what Wolfram Alpha does, for example - it manipulates expressions, not functions.

In the second setting, is not always obvious which function is being defined by an expression. For example, it is not at all obvious what $\int_1^3 zc$ would be - what function is $zc$?. In this case, we have to name the variable that is being used to define the function, for example $\int_1^3 zc\,dc$ means that we are integrating the function $f(x) = zx$ from $1$ to $3$, and per convention we are supposed to assume $z$ is a fixed real number. So the calculus formula $\int_1^3 zc\,dc = 4z$ really means: if $z$ is a fixed real number and $f$ is the function on $[1,3]$ with $f(x) = zx$ then $\int_1^3 f$ will equal $4z$.
Historically, nearer the time when calculus was developed, mathematicians such as Johann Bernoulli and Euler used "function" as a synonym for "expression". The routine use of functions that are not defined by particular expressions was not common at that time as it is now.
Modern authors are aware of the distinction, but for various reasons it is not emphasized in textbooks. Texts usually present integral rules in terms of expressions, but prove (or "prove") theorems in terms of functions. To bridge that gap, they sometimes need to indicate which variable is used to define a function from an expression. That is the modern purpose of the "dx" notation.