I've been learning about Quantum Dynamics, time evolution operators, etc. I am confused about the notation used in integrals. Normally I am used to integrals written in this way (with $dx$ on the right side): <br>
$$\int f(x)dx$$ <br>
In this manner of notation, I can easily see the integrand as it is sandwiched by the integral sign and the $dx$.
However, I often see integrals written in this way (with $dx$ beside the integral sign): 
$$\int dx f(x)$$
Is this notation not ambiguous? This is especially confusing for me when used in products, as I cannot identify what is the integrand sometimes. For example, I don't understand which is true in the following (when evaluating time evolution operator): $$\begin{align}\left(\int ^t_{t_0} dt' H(t')\right)^2\stackrel{?}{=}\int ^t_{t_0} H(t') dt'\int ^t_{t_0}  H(t'')dt''\\\stackrel{?}{=}\int ^t_{t_0} dt' H(t')\int ^t_{t_0} dt'' H(t'')\\\stackrel{?}{=}\int ^t_{t_0} dt'\int ^t_{t_0} dt'' H(t') H(t'') \end{align}$$

The last line is especially confusing for me as I'm not sure if the integrand changes. Could I please get clarification for these different notations? Is there a reason for such notation? (If I'm not wrong, it is to group the integrals and the integrands in separate places for convenience? I'm not sure if it sacrifices clarity for this though.)

EDIT: There is also an issue of when operators are involved:
$$\begin{align}\int dx \hat{F}(x) \hat{G}(x)\stackrel{?}{=}\int \hat{F}(x)dx\hat{G}(x)\\\stackrel{?}{=}\int \hat{F}(x)\hat{G}(x)dx\end{align}$$
How do you know which operator is in the integrand? And assuming the general case where $\hat{F}$ and $\hat{G}$ do not commute, you cannot write the integral with $\hat{G}(x)$ on the left of the integral. How is this not ambiguous?