You have told us nothing about ‘g(x)’. Assuming the integral of ‘g(x)’ is monotonically increasing at least until its integral is equal to 1 (if it’s periodic or has other pecularities, you will need another approach), try this:
x = linspace(1, 10); % ‘x’
y = rand(size(x)); % ‘g(x)’
vint = pi*sqrt(2)/3 * cumtrapz(x, y./x);
X_M = interp1(vint, x, 1);
Experiment to get the result you want.
