How to verify the correctness of following equation as Gaussian formula?

Question

How do I verify the correctness of this formula as Gaussian formula? $$ \int^1_{-1}f(x)dx= f(-\alpha)+f(\alpha) $$ $$\alpha=1/\sqrt{3}$$ I think I have to follow the theorem on Gaussian Quadrature, where we have $$ \int^b_aq(x)p(x)w(x)dx=0 $$ But im not sure how to find q, p, w

Answer 1

You know that an integral can be given by $$ \int_{\hat{x}}^{\bar{x}} f(x) dx = A_0f(\hat{x}) + A_1f(\bar{x}). $$

Hint: Check correctness of the formula for f(x), where f(x) is a polynomial of degree 0, 1, 2, 3, etc.