1. The obvious integral is (sin2q/2), and we recommend that you use this, because it leads to the stream function given by Eqn. (10.85).
2. But, from Appendix A, the integral is -(cos2q)/4, which is not the same, because if you expand the cos(2q), you get _1/4 + (sin2q)/2. However, this would be compatible with the answer in Item 1 above, because the _1/4 would cancel when limits of integration are inserted.