Past exam of the probability, statistics and options I course of the University of Oxford / 2011 / P / 2 / c / Solution

At either quadrature node, $2x_j^2=1$, so $u(x_j)=\arctan 1=\pi /4$. Thus the Gaussian quadrature approximation isThis is the requested quadrature value; the integrand is not a cubic polynomial, so the degree-three exactness established in part (b) does not make this the exact integral.