Solve with integrating an implicit function

[parisse]

1/ nSolve was added in Xcas for TI compatibility, but the native command is fsolve, nSolve differs from fsolve in one important aspect: it does not auto-quote argument, which is here important.
2/ I don't know why romberg works better than Gaussian quadratures... Perhaps this is again an evaluation problem
3/ Sign are indeed not opposite, I should have said check g(.7)-1/3 and g(1)-1/3, your definition of g is also correct.
4/ This is an evaluation problem, in algebraic notation you do not have enough control to make sure that the initial function f(x) will only be called with a numeric value for x. This can perhaps be improved by returning fsolve of the initial arguments.