How does one get from the left part of this equation to the center

integration by parts?
that's the first thing I'd try and check

>>7999783
>>7999888
Please be trolling

>>7999899
are you gonna say we should plug the infinitesimals ds and dt together and form v,
or is there a simpler derivation that's more rigorous?

>>7999936
I guess the more "rigorous" argument would be that you change integration variables

>>7999783
Is "s" supposed to be position?

>>7999947
given that v=ds/dt, the switch form ds to dv can't simply be achieve by an arbitrary Americanization of s.

>>7999953
Yes, it's always either x, r, l or s.
S is german for Strecke (meaning way or path).

>>7999783
[math]\int_0^s \!\frac{d(mv)}{dt}ds[/math]

[math]ds = \frac{ds}{dt}dt = vdt[/math]

[math]\int_0^s \!\frac{d(mv)}{dt}ds = \int_0^t \!v\frac{d(mv)}{dt}dt = \int_0^{mv} \!vd(mv)[/math]

>>8000799
That's a way, except the upper bounds on this are very odd then. What if |v(t)| isn't monotone increasing?
It doesn't really work as an integration variable.

I don't get it. OP where did you get that from?

>>8000799
Please explain the last part (with the definite integral from 0 to mv). How did we get there from the center integral?

This kills the mathematician. Seriously, who would use v in both the limits of integration and the integrand?

>>8001227
The same way we got vdt from ds in the step above.

>>7999783
If you had done applied math you would know how to do this