Dummy Variables and variables that change a funtion I am reading notes for my Calculus 2 class and am confused why these two functions are not equivalent
"
$x$ is not a dummy variable, for example,
$$\int 2xdx = x^2+C$$
and 
$$\int 2tdt = t^2 + C$$
are functions of different variables, so
they are not equal. 
"
Is it because the arbitrary C's can be different?

What we see here is an abuse of notation ingrained in our way of doing calculus since centuries. If ${\rm sq}:t\mapsto t^2$ is the squaring function on ${\mathbb R}$ nobody would denote the set of all functions differing from ${\rm sq}$ by an additive constant by $\{{\rm sq}+c\>|\>c\in{\mathbb R}\}$. Instead we all write $t^2+C$, or $x^2+C$, for this set of functions.
Similarly, given any function  $x\mapsto f(x)$ $(x\in I)$, e.g. $f(x):=2x$ $(x\in{\mathbb R})$, the typographical picture $\int f(x)\>dx$ by definition denotes the set of all functions $$t\mapsto F(t)\quad(t\in I)\tag{1}$$ satisfying $F'(x)=f(x)$ for all $x\in I$. We might as well have written $u\mapsto F(u)$ $(u\in I)$ in $(1)$. Indeed it is customary in integral tables to denote the independent variable on the right hand side again by $x$.
