I'm taking a course on ordinary differential equations, and in the last couple classes, my professor has written several integrals like this, with no lower limit: $$\int^xf(t)\ dt$$ From what I understand, he intends for it to be functionally the same as $\int f(x)\ dx$, so I'm wondering what the purpose of this notation is. Is it commonly used? (Our textbook doesn't use it, though.) Is it for some sort of clarity? Or is it perhaps just a quirk of his to write integrals like this?
Example:
In class today, we were talking about calculating integrating factors. At one point, we were working on solving the equation $xy'+2y=\sin x$ and he wrote this on the board:
$$\mu(x)=\exp\left[\int^x \frac2t \ dt\right]=e^{2\ln(x)}=x^2$$
I'm perfectly comfortable with finding integrating factors, but is there a reason not to write this without the $t$? It looks nicer to me as $$\exp\left[\int\frac2x\ dx\right]$$
When I have seen the notation $$\int^x f(t) \, dt,$$ it is used more for convenience than anything else. Here the upper limit of $x$ is used to remind one that the variable of integration needs to be changed back to $x$ after the integration with respect to $t$ has been performed.
In your particular case I think your professor is using such notation so as not to confuse the variable $x$ found in the integrating factor $\mu (x)$ with the variable used in the indefinite integral.