Thanks for the reminder, but I have already done so and it turns out they all goes to 0, so there must be something awry with my method. In particular, there seems always to be one spacial part of the integrand to be antisymmetric so that the whole integral goes to 0 (please refer to the image...
I have typed up the main problem in latex (see photo below)
It seems all such integrals evaluates to 0, but that is apparantly unreasonable for in classical mechanics such a free particle is with nonzero angular momentum with respect to y axis.