Should the notation $\int_{0}^{x} f(x) dx$ be frowned upon? In old mathematics books, I see a lot of notations like $\int_{0}^{x} f(x) dx$. For example, Courant-Hilbert: Methods of mathematical physics.
However, when I wrote it in this site, it was sometimes edited like $\int_{0}^{x} f(t) dt$. 
 A: This is wrong because it would not allow (without confusion) something like this:
$$\int_0^x f(t, x)\  dt$$
The variable of integration (or summation when doing sums) should always differ from all other variables because, to use expressions I recall (possibly incorrectly) from my youth,
the other variables are "bound" (the $x$ above) and the variable of integration is "free", so that the expression is unchanged if the variable is replaced by another.
For example, what would you make of this:
$$\int_0^x \int_0^x f(x,x)\ dx\ dx$$
instead of this:
$$\int_0^u \int_0^y f(x,y)\ dx\ dy$$
A: If you write $\int_0^x f(x)\,dx$, you have two different $x$s. One has scope inside of the integral, the other outside.  The term "scope" is somewhat strange in the mathematical world, but it means "where the variable has meaning."  
In this case it is far better to write $\int_0^x f(t)\,dt$; the variable $t$ is a "loop variable" or place-holder.
A: Over the summer I came up with an exercise for the kind of people who like to write $\int_0^x f(x) \, dx$: evaluate
$$\int_1^x \int_x^{x^2} \int_{x^2}^{x^3} x^4 x^5 x^6 \, dx \, dx \, dx.$$
I hope that my point is clear. 
A: Note that there are two different $x$'s in $\int_{0}^{x} f(x) dx$, which is made explicit when one is changed to $t$.  One is the upper limit of integration, which is still free, and the other is the dummy variable bound inside the integral. On careful reading one can tell them apart, but it is easier on the reader and less mistake-prone to distinguish them.
A: $\int_0^x f(t) \,dt$
is an expression that, by its limit definition, is basically the sum of an infinite number of areas of rectangular pieces of infinitesimally small width and height $f(t)$ for each value of $t$ between $0$ and $x$.  So, if we wrote
$\int_0^x f(x) \,dx$
instead, it would mean to add up the areas of these rectangles as the value of $x$ ranges from $0$ to $x$.  Hopefully, it is clear that this makes no sense.  $x$ can not simultaneously vary and stay constant.
