What is the purpose of the opamp in an integrator circuit?
Here comes another explanation: For integrating purposes we need a passive RC lowpass with a very low cut-off frequency. This would require excessive large capacitor values. 

Therefore - why not using the capacitance multiplication offered by the MILLER effect? Do you remember the MILLER effect in a simple transistor stage? Hence, we connect the integrating resistor to the inverting input of an amplifier which has a capacitor between inverting input and output. 
Now - the capacitor value is amplified by the open-loop gain of the opamp (as seen from the input). 

Example: 1nF*1E5=100µF. Together with an integrating resistor of R=10kOhm we would have a time constant of 1sec (cut-off at f=1/6.28 Hz). As another advantage - the output is available at the low-resistive opamp output. 

By the way: That is the reason the shown circuit is called "MILLER integrator".