Gaussian quadrature method when $n=2$. Function to be approximate is $\arcsin$, how to do it properly when weights are not given?

Question

So, there is a an exercise of numerical integration using Gaussian quadrature method. The givens are: f(x) = arcsin x, a = −1, b = 1, i = 2

I was just wondering, do I need to use the know formula for the $n=2$ to integrate the function, which is given as $f(\frac{-1}{\sqrt{3}} + f(\frac{1}{\sqrt{3}})$, because in that case the exercise seems to be trivial as $\arcsin$ is odd function and the integral would be equal to 0. What am I missing here? $$\int^1_{-1} f(x) = f(\frac{-1}{\sqrt{3}}) + f(\frac{1}{\sqrt{3}}) = 0$$ For this result the weights are chosen as $c_1=c_2=1$, but in the exercise that I need to solve the weights are not given, is this norm to just let $c_i = 1$?

score 1 · Accepted Answer · answered Jun 11 '20 at 23:29

Gaussian quadrature with weight function $w(x)=1$ on $[-1,1]$ is a well-defined thing, you don't need to be told what the weights and nodes are in order to do it. Specifically, the nodes are $\pm 1/\sqrt{3}$ and the weights are both $1$. Therefore, the answer to the exercise is indeed $\arcsin(-1/\sqrt{3})+\arcsin(1/\sqrt{3})=0$.

Gaussian quadrature method when $n=2$. Function to be approximate is $\arcsin$, how to do it properly when weights are not given?

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