Integrating Newton's First Law (F=ma) to obtain the expression for energy conservation for a book falling off a table
The force F in this case is constant, F=mg, m being the mass, g being the (constant) acceleration of gravity. So that F=ma is in this case gives:
mg = - m dv/dt
where v is the velocity (dv/dt being the "derivative" of v with respect to time, which is therefore the acceleration a), and the minus sign comes because gravity is pulling down and we are defining positive velocity as up.
Multiplying both sides by dt, and integrating both sides (choosing the simplest integration limits v1 = 0, v2= v, t1 = 0, t2=t) gives:
mg t = - m v
Replacing v with dh/dt (i.e. the vertical coordinate of the book is h - its height off the floor), and multiplying through by dt, we have:
mg t dt = - m dh
Integrating again (same limits for t as before, and h1 = htable, h2 = h) we have
mg 1/2 t2 = - m (h - htable)
From the first integration above (mgt = mv), we obtain t = - v/g (this equation tells us simply that v = - g t, i.e. the velocity increases downwards linearly with time, as it should under a constant force). Substituting this into the result of the second integration (to eliminate the explicit time dependence), we have
m g 1/2 v2 / g2 = - m (h - htable)
and multiplying through by g, and adding (m g h) to both sides, this yields our desired result:
m g h + 1/2 m v2 = m g htable (i.e. the energy equals the initial potential energy, which is obviously a constant)
Note that what made this all possible was the fact that we could eliminate the explicit time dependence - this is really a deep consequence of the symmetry of physical law with respect to time. If the force had depended on time explicitly, for example, obtaining a constant energy would not generally have been possible.