How to integrate current signal to be able to measure it with 50 ohm input impedance of ADC
You would use a non inverting op amp buffer between the RC integrator and the ADC.

That gives you the high impedance you need to connect to the integrator and the low impedance you need to connect to the ADC.

A (drastically simplified) example from the Wikipedia page looks like this:



The triangle shaped thing is the schematic representation of an operational amplifier.

The devil is, of course, in the details.

All opamps are not created equal:


Some have limited bandwidth.
Some have limited input voltage ranges.
Some are not stable when used as a buffer.
Some can drive low impedance loads better than others.
There are many more properties of opamps than I've listed here.


Building a completed buffer around an opamp and meeting particular requirements of bandwidth, noise rejection, voltage input range, etc. can be challenging.

If your ADC is a circuit you have built yourself, then you may want to learn about opamps and build your own buffer.

If you are using a commercial ADC module (data acquisition system of some kind) then you will probably want to consider using a commercially made buffer module.  Find one that meets your needs for bandwidth, voltage range, noise, what-ever-else, then pay your bucks and get on with your measurements.

You might also look into ADC modules with high impedance input.  It might be cheaper to use an appropriate module to start with rather than trying to continue with a piece of equipment that doesn't do what you need.