Apply gaussian quadrature with n = 4 to approximate integrate sin x^2dx from 1 to 5

First, recall that Gaussian quadrature is based around integrating a function over the interval [-1,1], so transform the function argument accordingly to change the integral over [1,5] to an equivalent one over [-1,1].





So,



Let . With , we're looking for coefficients and nodes , with , such that



You can either try solving for each with the help of a calculator, or look up the values of the weights and nodes (they're extensively tabulated, and I'll include a link to one such reference).

Using the quadrature, we then have