Re: v-f lm331
U1A is used exactly as suggested in the datasheet of the LM331 (page 8). It is an integrator, which improves the accuracy of the converter. The integration being done by the opamp, the voltage at pin 1 is maintained close to GND. Plus, you can select opamps with very low offset and input bias currents. All that means better linearity.
U1B is there simply to provide a high impedance to the input voltage and a low impedance at pin 1 of R5.
In additon, by vaying the gain of U1B (by changing R1 or R2) you can adjust the full-scale frequency even if your input voltage has a limited range. For example, let's assume the input voltage can only change from -1 to +1V. But you need -2V to +2V to cover the required frequency range. So you can just make the gain of U1A equal to 2 and "presto"! You cover the range without having to change the input voltage range (which may be impossible).