Example 14.2 of Concepts in Thermal Phyiscs, 2nd ed:
"Consider two systems, with pressures $p_1$ and $p_2$ and tempreatures $T_1$ and $T_2$. If internal energy $\Delta U$ is trasferred from system 1 to system 2, and volume $\Delta V$ is transferred from system 1 to system 2, find the change of entropy. Show that equilibrium results when $T_1 = T_2$ and $p_1 = p_2$."
The given solution is to use the first law of thermodynamics, $dU = TdS - pdV$, and rewrite it as $dS = (1/T)dU + (p/T)dV$. Then the change in total entropy $\Delta S = \Delta S_1 + \Delta S_2$, is "straightforwardly":
$\Delta S = (1/T_1 - 1/T_2) \Delta U + (p_1/T_1 - p_2/T_2) \Delta V$.
So if $dz = (1/y)dx$ then $z_2 - z_1 = \int dz = \int (1/y)dx = (1/y_2 - 1/y_1)(x_2 - x_1)$. Why? The last step is not at all straightforward to me. Certainly it's not true in general. Nevermind, I confused the meaning of indexes 1 and 2; they denote systems 1 and 2, not "before and after".