Is it possible to make an op amp integrator that integrates a DC input?
Yes, the textbook integrator circuit will output a ramp if you feed it a DC voltage. The inverting integrator is easiest. 







simulate this circuit &ndash; Schematic created using CircuitLab

Of course you have to establish an initial condition (for example, by shorting the capacitor) before you want the ramp to start. At some point the integrator will hit a limit (usually when the output approaches a supply rail) and the integration will stop.  

In the above circuit the voltage will (ideally) be 0V just after the switch opens and and will ramp upward at 1 volt per second. 

The op-amp inverting input is at a virtual ground (so long as it is balanced) so there is 1V across R1, and 1uA flows. The input ideally acts as an open circuit, so there must be 1uA flowing through the capacitor and thus dv/dt is 1V/second. When the output hits the (positive in this case) rail or thereabouts, the output will stop changing so the inverting input will no longer be at 0V- it will charge exponentially towards -1.0V with a time constant of 1 second.