Two-node Gaussian quadrature with sine weight ($c_{1,2}=\pi /2\mp \sqrt{{\pi }^2/4-2},b_1=b_2=1$)

For integration against $\sin x$ on $[0,\pi ]$, the monic quadratic orthogonal polynomial is $x^2-\pi x+2$. Its two roots are the Gaussian quadrature nodes $c_{1,2}=\pi /2\mp \sqrt{{\pi }^2/4-2}$, with weights one. The rule integrates every cubic exactly: divide it by the orthogonal quadratic and integrate the degree-one remainder.