Good morning to all, this morning (ah.. European morning) I gave a calculus exam and one of the problems I could not find an answer to was...

Which of the Gaussian quadrature forms would you apply to solve the following integral:

$
\int_0^1 \sqrt{x(1-x)}P_7(x)dx
$

Where of course the P(x) has a degree n=7.

I thought about using the Jacobi polynomials, assuming that the weight function was

$
w(x) = \sqrt{x(1-x)}
$

but I am not sure if this is correct, it should be more like (1+x)(1-x).

What approach should have I followed?