Taking (numerical) integral over a function's implicit inverse - Matlab implementation
This function is to take integral over a function which is monotonic over the interval of interest but its inverse has no analytical solution (or the solution is too messy and is not worth finding). Legendre-Gauss Quadrature is used to estimate the integral numerically. It is possible to use a weighting function, for example when this function is weighted by a distribution function PDF such as Normal, Log Normal, etc. To avoid such weights use @(x) 1 as weighting function.
To find L-G quadrature nodes you need the file "lgwt.m", a function written by Greg von Winckel which I downloaded from: https://www.mathworks.com/matlabcentral/fileexchange/4540-legendre-gauss-quadrature-weights-and-nodes
Note: I have written this subroutine (and many more) because I frequently use it in part of my own research. Use this at your own risk.