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I think the answer is easier than you realise.
Lets do it step by step.
1) Suppose we have velocity v0 at time 0, and velocity v at time t.
Now we want velocity to vary from v0 to v, so the book uses v' as a
'dummy variable', which represents velocity as a variable when v is
fixed for that particular instant t. v' is what varies from v0 to v. It
does not mean the derivative of v! It is simply a formal notational
issue. We may allow the limit v of the integral to change, but this
causes the entire integral to change value -- ie v becomes a variable
of which the integral is a function. We only need to think of v as
fixed When We Are In The Process of Integrating... this is where the
dummy v' comes into play. In practice, such formal notational points
are often neglected.
2) Note that 1/(dv'/dt)=dt/dv'.
3) Then 1/f(v') = 1/m * dt/dv'.
4) Integrating, int(v0 to v) {1/f} dv' =
1/m * int(v0 to v) {dt/dv'} dv'
Here we make a change of variable to t, giving 1/m * int(0 to t) dt =
t/m.
I hope this answers your question.
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