I am working on studying for my last Qualifying Exam and ran across a problem that has stumped me. I have seen someone else's work for this, but I am pretty it is wrong. Anyway...

The solution I have seen is as followsLet be a smooth function and consider the integral

and consider that interpolates at the roots of the Legendre polynomial of degree n.

Use to derive a quadrature method for the integral

So my issue is that he didn't really use Legendre polynomials. Instead he bipassed the Legendre polynomials in favor of a Lagrange polynomial that interpolates at the Legendre roots. Maybe I am missing something or just confused. Could someone help me either understand the work above or help me with the problem? Thanks in advanced.Let be the distinct roots of (the Legendre Polynomial of degree )

Define