Substitution theorem with singular Jacobian matrix

Hi! I'm trying to compute with Matlab (Gauss quadrature) an integral which results from the application of the substitution theorem. So, inside the integral, I have a term which is the Jacobian matrix of a transformation function.

To perform the integration, I have to invert the Jacobian matrix and then to integrate the whole integrand. Unfortunately, it seems that in some integration points the Jacobian matrix is singular, and therefore not invertible.

Why does this happens and how do I compute the integral if I cannot compute the integrand in that integration point? Is there any solution for this?

Thanks!