Formalism in integration Let's say we have some $y(t)$. The derivative of $y$ along time axis will be $y'(t)=\frac{dy(t)}{dt}=\frac{dy}{dt}$. So I will integrate like this over time:
$\require{cancel}$
$\int_{t=0}^{+\infty}\frac{dy}{\cancel{d\tau}}\cancel{d\tau} = \int_{t=0}^{+\infty}dy=y|_{t=0}^{+\infty}=y(\infty)-y(0)$ , (since $y = y(t)$)
Even though the result being correct, is this procedure correct according to formalism?
In other words:


*

*can I cancel the differentials? Thinking of interval that makes sense to me.

*can the integral variable be different from the differential (second step)? I've almost never see the integration variable explicitly shown like I put there.

*I solved the integral in the third step as my variable was $y$ even though it was actually $\tau$. Can that be done?


Since you are asking about being formal, many would insist on using limits when doing an improper integral:
$$
\int_a^\infty y(t) \; dt = \lim_{K \to \infty}\int_{a}^K y(t)\; dt.
$$
Note that $y(\infty)$ doesn't (formally) make sense unless you have a function where $\infty$ is an element in the domain.
I would also say that by writing:
$$
\frac{dy}{\cancel{d\tau}}\cancel{d\tau}$$
you are implicitly saying that the derivative is a fraction and many would object to this (in particular in a basic calculus class). I, by the way, assume that $\tau$ is the same as $t$ in your example. Otherwise there is something that I don't understand.
