How is the integration time set for a camera?
It's a crappy datasheet. As you had noted before:


  According to the data-sheet the integration time is supposed to go between the "SI (Start Integration) positive pulse and the HOLD positive pulse". At the same time it says "the SI and HOLD pins tied together" so I'm confused


It is written as if the Hold signal is on the pin but in reality (from page 2:)



It is clearly an internally generated signal.

What you're probably not getting is that the analog read out is the data from the previous integration period.  The device is integrating the newest image whilst you are reading out the previous one.

You can see this from the timing diagram. I've added in a red arrow showing that the "integrating" just under the \$t_{int}\$ follows in the next line period.



and this is apparent from the pixel structure as well. There is an integration capacitor as well as a hold capacitor.



SO the key sentence you should care about is this: (from page 2)



What that means is that to get longer integration times, you will need to stop the clock and wait.  That means that your line rate will go up as well, but if you are running at max line rate then you are stuck with a fixed integration period.

Of note is that you may need to stop the clock at the end of the read out. i.e. extending \$t_{qt}\$.  OR you may be able to keep it free running and then start the next sequence with the SI pulse.  Dig through the data-sheet with your new knowledge or experiment.

The internal sequence of events inside the chip is:


On the SI pulse, 1st transfer the recently integrated values into hold capacitors, and start ready out in sequence from pixel one throughN.  In parallel reset the pixels ready for a new integration. This resetting takes 18 clock cycles.


After 18 clock cycles start integrating. This integration then happens in parallel with the presently ongoing readout (from pixel 19 onwards).


Once the last pixel clocks out you have to wait \$t_{qt}\$ until hitting SI again.  At this stage all the old pixels are read out but the integration is still happening.  If you delay the start of SI then the integration extends. 


NOTE: that the end of the integration ends with the rising edge of SI.