Why is $\int_0^x f(x) dx$ wrong? As soon as I entered higher education we stopped using functions of $x$ as the upper limit if we were integrating using a function application on $x$ like $f(x)$.
So we were told that an integral of the form:
$$\int_0^x f(x) dx$$
Was considered incorrect, why is this technically incorrect?
Is it due to the 'bound' occurrences of $x$? Or some other reason? Why do we tend to introduce a 'dummy' variable like the following:
$$\int_0^x f(s) ds$$
In this case, is the bound variable's 'scope' limited to the integral, can I take a function's value at a 'free' occurrence $x$ and also use it in the integral, and have it keep it as the same variable?
 A: Purely technically speaking, it is not wrong. But it is very bad form, and can lead to a lot of confusion quickly. The confusion comes from two concepts clashing with each other. That is, in the expression
$$\int_0^x f(s)ds,$$ it is immediatelly clear that $x$ is a free variable, and $s$ is not a free variable. In other words, the question

What is the value of the expression $\int_0^x f(s)ds$ when $x$ equals $3$?

is clearly a well defined question with a single correct answer. Even more importantly, that answer can easily be obtained by just replacing all instances of $x$ with $3$. On the other hand, the question

What is the value of the expression $\int_0^x f(s)ds$ when $s$ equals $3$?

is clearly an ill posed question.

In your case, however, some of the instances of $x$ are free variable, while others are not. In particular, the $x$ in $f(x)$ and the $x$ in $dx$ is not a free variable. Therefore, the question

What is the value of the expression $\int_0^x f(x)dx$ when $x$ equals $3$?

is still a well posed question, and the answer is $$\int_0^3 f(x)dx$$ which can be obtained by replacing all cases where $x$ is a free variable with $3$.
However, and this is the important part, this answer can now no longer be obtained by simply replacing all instances where $x$ appears with $3$. And this can become very painful very soon. For example, say we have another function, $g(x)$. Then, we want to multiply your expression with $g(x)$. Well, what we get is $$g(x)\int_0^x f(x)dx$$ but then what we cannot do is say that that is equal to $$\int_0^x f(x)g(x) dx,$$ but we could have said that $$g(x)\int_0^x f(s)ds= \int_0^x g(x) f(s)ds.$$ In other words, the notation is constraining our ability to manipulate the expression.
