How to discharge a capacitor being charge by an op-amp?
This is problem that generally occurs whenever you implement an analog integrator. 

Analog switch across the capacitor, or a relay (with a small resistance in series). Ideally stop the current into the amplifier so the switch resistance does not compromise the reset. 

For example, here is an analog switch suitable for +/-5V supplies that is less than 6\$\Omega\$ when on, and incorporates level shifting. 



There are various analog switches available, some with logic level shifting, some without, some will tolerate 5V or less, some +/-15V. Charge injection (typically in the pC range) may be important if your capacitor is relatively small because the charge injection will affect the capacitor voltage when you open the switch. Generally speaking, the lower voltage your supply rails, the lower switch resistance is possible.

Another possibility is to close the loop around the integrator and drive the output voltage to zero, but you're probably looking at analog switches again. This is done in some meter chips. 

Edit: 

This is what an analog switch looks like internally, two MOSFETs with inverse drive: 



Note that the body connections are not to the source as usual for discrete MOSFETs.