# Peculiar way of writing limits of integration

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 $$\exp\left[\int\frac2x\ dx\right]$$

• Why haven't you asked your professor this question??
– user296602
Commented Jan 22, 2018 at 21:32
• People have asked him in class multiple times, and he has yet to give a real answer. I believe he's said the book uses it (which it doesn't), and he's also said that it makes things simpler, which I don't understand. I'm just wondering if anyone else has seen or used this notation. Commented Jan 22, 2018 at 21:35
• I think the notation probably means the integral is still a definite integral but the lower bound can be chosen conveniently to make the result simple. For example, in $\mu(x)$, the lower bound for $t$ is implicitly chosen to be $1$ to cancel any constant. Commented Jan 22, 2018 at 21:36
• I had not seen that notation before, I think is just a notation used for your teacher. It seems he don't care about the constant appearing when integrating, that make sense in ODE context. But I think that here we only can speculate. Commented Jan 22, 2018 at 21:39
• @Zhuoran He That makes sense; thanks! Commented Jan 22, 2018 at 21:48

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.