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.

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.

Edit: Note the comment by @G36. It does look like the collector and emitter are swapped. That would possibly stop it operating.

Note that it might not affect operation, BJTs can operate with collector and emitter swapped but they will have very low Hfe and the base emitter breakdown (now base-collector if swapped) is only about 7V as mentioned earlier.

The base drive from the second opamp will pull the inverting input of opamp1 to a low level through the broken down base-emitter junction. The low level will cause the output to incorrectly be at 11v.

In future when posing questions with schematics please make sure that they have component references, otherwise it makes it very difficult to explain which component is being referred to in an answer.