Sometimes I see $\int$ there but no $d$ follows. Why? In my opinion, the mathematical symbols $\int$ and $d$ should always appear together to represent an integral form, such as 
\begin{align}
\int_{x_0}^{x_1} e^x dx = e^{x_1}-e^{x_0}
\end{align}
In the LHS, $\int_{x_0}^{x_1}$ and $dx$ construct an integral for the variable of integration $x$ over $[x_0,x_1]$. However, I do see the integral sign $\int$ alone without $d$ sometimes in some math formula, for example
\begin{align}
\int_{\partial \Omega} \omega = \int_\Omega d\omega
\end{align}
The LHS of this classic Stokes' formula does not contain $d$. Why the $\int$ can appear without $d$? I do observe many other cases like this, typically in real analysis and differential geometry. Do they just omit $d$ and consider the variable of integration in default? 
 A: To put it simply (and maybe a little harshly), your opinion is wrong.
Given a function $f:[a,b] \to \mathbb{R}$ (let's stay in intervals just to simplify the discussion), the symbol $\int_{[a,b]}f$ is completely well-defined (be it Riemann, Lebesgue or whatever integration theory you are using). For instance, your first equation is
$$\int_{x_0}^{x_1}\exp=\exp(x_1)-\exp(x_0).$$
In fact, it is arguably more accurate than its "brother" $\int_{[a,b]} f(x) dx$, which props up in things like $\int_{[a,b]} x^2dx$ for example. Such notation is mostly used for computational (e.g. applications of Fubini, change of variables etc) and pedagogical purposes, because otherwise we would need to write
$$\int_{[a,b]} (x \mapsto x^2).$$
instead of
$$\int_{[a,b]}x^2dx$$
among other things.
The "$d$" in Stokes's theorem is another thing altogether, and its conception is much more elaborate than some texts/people imply by not even wrongly saying
that it formalizes/generalizes the "dx" appearing in integration (which actually doesn't appear and is a notational fantasy).
