## Change of Variables

I asked this in another thread, but I think this forum might be a better place for it (not trying to spam the same question). When deriving the formula for relativistic kinetic energy, we start with
$$KE = \int_{0}^{s} \frac{d(mv)}{dt} ds = \int_{0}^{mv} v d(mv)$$
So I figure that since v = ds/dt, then the right side must come from:
$$\int_{0}^{s} \frac{d(mv)}{dt} ds = \int_{0}^{mv} \frac{ds}{dt} d(mv)$$
I'm having trouble seeing how to change the limits of integration in a situation like this. The 's' just became 'mv', but what if it had been something more complicated than just 's', like say, tan(s) or something. I'm trying to understand this process. If it had just been a normal substitution u = g(x), then I would understand how to change the limits, but this seems like something different.
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