That circuit should work.
You need to measure some voltages to determine the cause of the problem.
That is half the job of being an engineer, to understand your circuit well enough and do the required measurements to correct any issues.
The transistor should be non-conducting with -10V on its base so the integrator should be giving out +11V not -11V.
One undesirable point with the circuit is that when operating from 12v the negative voltage to the base of the transistor will exceed its breakdown voltage (typically about -7V). This should not stop the circuit working but can cause degradation of the transistor.