Hey Tommy,

I'm not on this site so often; I hope this thread isn't too cold that posting suggestions here is still worth while.

The function in the integral is a weight function. Notice that it is positive everywhere on the interval .

First you need to find the first few orthogonal polynomials for this weight, that is, polynomials satisfying

,

and satisfying . You can do this by applying Gram-Schmidt to the polynomial basis . You can also do this by hand, since it only involves definite integrals of polynomials.

The quadrature rule you want has three "abscissae" , which will be zeros of the polynomial that you find. This will give a quadrature rule of maximum precision.

Note:. If not, you made a mistake somewhere!

It does not matter if you normalize the when calculating. Scaling a polynomial by a constant does not affect the location of its zeros. However the coefficients

are relevant to the weights in the quadrature rule. There is a formula; also for ; if you want, I can look around and maybe find a reference online.

Yours,

Jerry