How to distinguish between explicit substitution (u-sub) and implicit substitution?

Question

$$u=2x+1\tag{1}$$

$$x=a\sec\theta\tag{2}$$

I know that $(1)$ is an explicit substitution (u-sub) and $(2)$ is an implicit substitution. I know this because a person told me this. I couldn't have figured it out on my own.

Questions:

1. How do I distinguish between explicit substitution (u-sub) and implicit substitution in the future?
2. Do all implicit substitutions need to be invertible?

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Related: 1, 2

Answer 1

In the context of integrating $$\int f(x)\,\mathrm dx,$$ making an explicit substitution means to make a substitution of the form $$v=g(x)$$ (I'm deliberately avoiding using $u$ to draw attention away from its name), where the new variable $v$ is expressed as a function of the original variable. On the other hand, an implicit substitution is of the form $$x=h(v)\quad\text{or}\quad h_1(x)=h_2(v),$$ where the new variable $v$ is the argument of some function in the substitution.