Op_Amp as an integrator
This is a more conventional design of an amplifier.
Note the compensation capacitor, C1 which gives the amplifier stability by providing adequate phase and gain margins. - It rolls off the open loop gain, with increasing frequency, so that the loop gain gets down to unity well before the loop phase reaches -360 degrees. Typical value for C1 would be 100pF.
You can replace Q8, and its associated components, with your two biasing voltage sources if you wish.

The integration process on its own produces a negative cosine wave (90 degrees lag) but the op amp integrator is itself inverting, inverting the negative cosine wave to give an output cosine wave with 90 degrees lead on the input sine wave.