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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?

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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 $2n-1=5$

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  • $\begingroup$ So can we determine the Gaussian quadrature formula only at $[-1,1]$ ? Which is the weight function in this case? $\endgroup$ – Mary Star Feb 5 at 19:45
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    $\begingroup$ The coefficients $c_i$ and the nodes $t_i$ are listed in numerical texts. Once you made the transformation to $t$ you apply the quadrature formula to approximate the integral. $\endgroup$ – Mohammad Riazi-Kermani Feb 6 at 0:27
  • $\begingroup$ But for the calculations of $c_i$ and $t_i$ do we not need the weight function? $\endgroup$ – Mary Star Feb 6 at 20:08
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    $\begingroup$ You need $ f(t_i)$ and $c_i$ to find the quadrature. $\endgroup$ – Mohammad Riazi-Kermani Feb 6 at 20:23
  • $\begingroup$ I got stuck right now. In general we have an integral $\int_{-1}^1f(x)\cdot w(x)\, dx$, or not If we have $w(x)=1$ do we consider the Legendre polynomials? $\endgroup$ – Mary Star Feb 6 at 20:32

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