Collecting Sample points on the oscilloscope

There is a significant but easily fixable error in your circuit. You have the negative supply connected to 0V and also your op-amp's non-invering input at 0V. This means for positively charged particles hitting the cup you'll see nothing from the op-amp because its output will want to go below 0V (but it can't). Definitely try symettrical supplies such as + and - 15 volts.

A side effect of the bad negative rail connection is that a lot of op-amps will just go belly-up and produce seemingly daft output voltages. This is op-amp dependent of course.

Another issue and this may show up is the input bias currents needed by the op-amp. Just any old op-amp will not work - you have to choose an op-amp that has massively low input bias currents or the leakage through the input protection resistance will produce a huge offset voltage.

Back in the 1990s I designed an ion beam collector for a gas mass spectrometer (that worked of course!) and we used the damned-best input leakage current op-amps we could find. ADI produce one that is in the order of several fempto amps and doesn't cost a king's ransome. I can't remember the part number now but just google "ADI fempto amp op amp" and you should get a hit.

I'd also consider not using an integrator as the front-end. I'd use a fempto-amp op-amp with the 100Mohm resistor in the feedback, dispence with the input protection (unless you have very good reasons to keep it) and have this followed by an integrator then, in that 2nd stage you can use a reed relay or JFET for resetting the capacitor should you need to. It's quite a valid approach to smaple then reset, let build up, then sample then reset continually. If the reset is short it won't significantly affect things but will give you the ability not to crash the output into the power rails if too much charge is built up.

There are different ways to trigger your scope.

1. Use your scope in single-shot operation. Trigger once then work on the captured waveform.
2. Use the math functionality (page 6-21) with a low-pass filter to filter the PWM signal. With the proper cutoff-frequency you should be able to see the sinewave. You could display the PWM on another channel and overlay the PWM with the filtered signal. You should be able to trigger on the math channel. Update: Triggering on the math channel is not possible with this scope.
3. Use an external trigger. The scope has a trigger input that you can use. You could program your microcontroller to generate the trigger signal on another pin that you can feed to your scope (see chapter 5).
4. Filter the PWM using a two-stage RC-filter and trigger on the sinewave. Use two channels the see the PWM and the filtered signal.

Actually you should try all of these since this a nice exercise that will help you to get to know your scope better.