What I know, any integral can be transformed into a different function and weighted factor in Gaussian-Quadreture method. The the limit is [-1, 1].
But let assume we have an integral, $$f(x) = \frac{1}{1-x}$$
If we change this integral to a Gaussian Quadrature, how will I deal the the value of x=1, I'm saying this because for, x=1 we get undetermined form. Even we change the integral, where does the improper portion go?
I mean, I f we change any integral to Gaussian Quadrature integral form, how do we know that we aren't dealing with the pole of the integral anymore?
Take a look at a table of Gaussian quadrature abscissas. Do you see $x = 1$ in any of them?
In fact it's a theorem that the abscissas of the Gauss-Legendre quadrature are the zeros of the Legendre polynomials, which lie strictly in the region $(-1, 1)$.
So endpoint singularities do not invalidate Gaussian quadrature rules, though they can become inaccurate. A bigger deal is that $\frac{1}{1-x}$ is poorly approximated by polynomials, so Gaussian quadrature is a poor choice.