Help understanding integral substitution I don't quite understand the steps involved in this indefinite integral with respect to $t$:
$$\int\frac1{M}\frac{dM}{dt}$$
The explanation I've does this substitution:
$$u = M(t)$$
Which seems to make the integral become:
$$\int\frac1{u}du$$
From here I understand that we integrate with the exponential rule, to get:
$$\ln|u| + C$$
My explanation then substitutes back $M$ to get:
$$\ln|M| + C$$
These are my understanding problems:

*

*What is the meaning of integrating $dM/dt$? I'm familiar with integrating functions with respect to one variable, like this: $$\int f(x)dx$$ but $dM/dt$ is the notation for a derivative? I just don't understand what we're doing here.


*I don't quite get why the substitution $$u = M(t)$$ works here. Probably relates to understanding problem 1.


*Why when we substitute $M(t)$ back we get: $$\ln|M| + C$$ instead of:
$$\ln|M(t)| + C$$
the later looks strange, but we did $u = M(t)$, so my primitive understanding tells me to substitute back literally $M(t)$. Is the $(t)$ implicit?
 A: Your confusion is understandable.  The notation really should be written $$\int \frac{1}{M(t)} \frac{dM}{dt} \, \color{red}{dt}. \tag{1}$$  The reason becomes apparent if we pick a specific example; e.g., $$M(t) = \sin t.$$  Then the integral $(1)$ takes on the form
$$\int \frac{1}{\sin t} \cos t \, \color{red}{dt}. \tag{2}$$  Without the $dt$ (in red), the original notation would read
$$\int \frac{1}{M} \frac{dM}{dt} = \int \frac{1}{\sin t} \cos t$$
and this is not really complete.
As for how the substitution works, you can also see it through the example I chose:  we let
$$u = \sin t, \quad du = \cos t \, dt, \tag{3}$$ which gives
$$\int \frac{1}{\sin t} \cos t \, dt = \int \frac{1}{u} \, du = \log |u| + C = \log |\sin t| + C. \tag{4}$$
The same mechanism is valid for a generic differentiable function $M$:
$$u = M(t), \quad du = M'(t) \, dt = \frac{dM}{dt} \, dt. \tag{5}$$
The confusion over whether to use $M(t)$ or $M$ is more of a matter of notational brevity:  when $M$ is used, it is implied through the term $dM/dt$ that $M$ is some function of $t$.  Either $M$ or $M(t)$ is acceptable.
