I want to calculate the integral $$\int_0^1\frac{1}{x+3}\, dx$$ with the Gaussian quadrature formula that integrates exactly all polynomials of degree $6$.
First of all do we have to transform the intervall of the integral, i.e. to shift it from $[0,1]$ to $[-1,1]$, or not?
Then I thought that it holds the following:
The gaussian quadrature integrates exactly polynomials $\Phi (x)$ with maximum degree $2n-1$. $2n-1$ is an odd degree but we have in this case the degree $6$, which is even.
What do we have to do in this case? Or is the way I am thinking wrong?
The answer to your first question is yes, you have to transfer the interval to $[-1,1]$ in order to apply the Gaussian Quadrature. The transformation $x=\frac {t+1}{2}$ would do the trick.
For your second question you need to consider $n=4$ so you get $2n-1=7$ which covers the polynomials of degree 7 as well as polynomials of degree 6. Clearly $n=3$ is not sufficient because you get $2n-1=5$