Operational Amplifier Circuit
For reference and to protect against future edits, here is the circuit we are discussing:



This is simple to analyze if you break it into its two distinct parts.  You left out component designators in the schematic, so it will be difficult to talk about.  You'll just have to guess which ones I mean.  Next time, draw the schematic properly.

The IC1 is just a classic integrator.  Due to the feedback, the opamp does what it needs to so that the - input is kept at 0 V.  This means the current flowing thru R1 is directly proportional to Vin.  The only place that current can go is thru C1.  By the nature of what a capacitor does, the voltage across C1 is the integral of the current thru it.

The IC2 is just a classic inverting amplifier.  Ignore the V1 input for now.  The gain from the output of IC1 is simply -R3/R2.  Note that IC1 actually inverts while integrating.  IC2 inverts the signal again, so Vout is the positive integral of Vin.

V1 is just another input added to that from IC1.  The gain from V1 to Vout is -R3/R4.  This is simply added to the previous result.  In that sense, you can think of IC2 as a summing amplifier.