Why is it considered incorrect to use the variable of integration as a boundary of integration? Recently, a question was asked about limits of integration and the OP made the following "typo":
$$I =  \int_0^x f(x) \, dx$$
as opposed to
$$I =  \int_0^x f(t) \, dt $$
I commented that this typically frowned upon and can make for some odd results. It occured to me however, that I wasn't sure why this was formally incorrect, but that I had just been taught that it shouldn't be done. I couldn't think of a reason why it's considered best practices to differentiate between variable of integration and variables in the limits of integration..

Any thoughts / reasoning behind this?
Let's consider a simple case:
$\int_0^x x\; dx$.

Applying the usual method,
$\int_0^x x\; dx
= \dfrac{x^2}{2}\big|_0^x
=\text{?}
$

The lower term is
$\dfrac{x^2}{2}$
evaluated at $x=0$,
and this is obviously $0$.

The upper term is
more problematical.
It is
$\dfrac{x^2}{2}$
evaluated at $x=x$,
and it is unclear what this means.

Suppose the integral was
$\int_0^{2x} x\; dx$.
The upper term would then be
$\dfrac{x^2}{2}$
evaluated at $x=2x$,
and it is even more unclear what this means.

Good notation makes ideas clear
and helps avoid mistakes;
as shown here,
using the same variable for
the variable of integration
and a limit of integration
can make things unclear.
In addition to what Marty said i think the latter integral (too lazy to type!) is considered wrong mainly because the limits of integration are a function of x as well.
A big contradiction to the equality of former integral and latter integral is when you differentiate the integral. 
Just try that thing out in both the cases we get totally different answers. 
latter integral is a f(x,t) which has a different and more complicated differentiating form. 

There might be many more reasons to this but yeah you should get the idea. 

If however the limits are independent of x then you can always do that x --> t shift .