Consider the quadrature given by

$$\int_{0}^{\pi} w(x) f(x) d x \approx \sum_{k=1}^{\nu} b_{k} f\left(c_{k}\right)$$

for $\nu \in \mathbb{N}$, disjoint $c_{k} \in(0, \pi)$ and $w>0$. Show that it is not possible to make this quadrature exact for all polynomials of order $2 \nu$.

For the case that $\nu=2$ and $w(x)=\sin x$, by considering orthogonal polynomials find suitable $b_{k}$ and $c_{k}$ that make the quadrature exact on cubic polynomials.

[Hint: $\int_{0}^{\pi} x^{2} \sin x d x=\pi^{2}-4$ and $\int_{0}^{\pi} x^{3} \sin x d x=\pi^{3}-6 \pi .$ ]