How do I determine the weights and abscissas in the 1 and 2-point Gauss quadrature given a weight function?

Question

Determine the weights and abscissas in the 1 and 2-point Gauss quadrature formulae for $\int_{0}^1 f(x)w(x)dx$ with weight $w(x) = − \ln x$.

I'm pretty confused on how to approach this problem with a given weight function

Answer 1

The one point formula would be $$\int_0^1 f(x)w(x)\approx a_0f(x_0)$$ where you need to choose $a_0,x_0$ to make the integral exact for polynomials of as many degrees as possible. Start with constants, so let $f(x)=1$. Do the integral and choose $a_0$ to make it correct. Now do a first order polynomial, so let $f(x)=x$. Again, do the integral, and choose $x_0$ so the approximation is correct. You are out of adjustable parameters, but it is possible that you are lucky and it works for quadratics as well, so try $f(x)=x^2$ and see.

For the two point formula you have two coefficients and two points to choose. You should be able to get the formula exact through cubics because you have four adjustable parameters. The process is the same, but you will have to solve some simultaneous equations.