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.