I'm reading through a modern physics text (on my own) and it did some very strange manipulation when deriving the relativistic form of the kinetic energy equation. It starts off deriving relativistic force which I understand. Basically, the implicit derivative of (d/dt)(ymv). Y is gamma and is equal to the Lorentz ratio 1/sqr(1-v^2/c^2). (I do not know how to write the notation.) Work or kinetic energy(if no work goes to potential energy) is equal to the integral from 0 to s of Fds (Must know how to write notation of this site.) With substitution we get the integral from 0 to s of (d/dt)(ymv)ds. This is where the book does some strange things. The author claims that the integral for 0 to s of (d/dt)(ymv)ds is equal to the integral from 0 to mv of v*d(ymv), and once y is substituted in, the integral is equal to the integral from 0 to v of v*d(mv/sqr(1-v^2/c^2)). He has changed the bounds of his integration twice in a way that I do not understand! Does anyone know how he changed the bounds accordingly as he substituted variables in? I fallowed the rest of his work. He does integration by parts simplifying things greatly.