gaussian integration

Question

I'm new on Mathematica and I am an engeneer with only a little base of numerical computation. I have to integrate a trigonometrical function numerically with a Gauss integration. The function is:

   t[alpha_,x_]= ( 2 Sin[alpha]^4 - Cos[alpha]^2 Sin[alpha]^2 ) 1/(
 2 Sin[alpha]^4 (Cot[alpha]^2 - 1) (Cot[alpha]^2 - 
    0.5)) (-(1/(Cos[alpha]^2 Abs[x]^2)))

I have to integrate it in the interval from 0 to Pi. Then I have to find couples of points and weights like this

f = GaussianQuadratureWeights[n, 0, Pi]

with n the number of point and after add weight for the function t evaluated in the point?

    g = 0
For[ i = 1, i < n + 1, i++,

 g = g + f[[i, 2]] t[f[[i, 1]],x]
 ]

Answer 1

 g[a_] := ( 2 Sin[a]^4 - Cos[a]^2 Sin[a]^2 ) 1/(
   2 Sin[a]^4 (Cot[a]^2 - 1) (Cot[a]^2 - 0.5)) (-(1/(Cos[a]^2) ));

GaussLegendreQuadrature[a_, b_, n_, f_] := 
 Module[{weights, i}, weights = GaussianQuadratureWeights[n, -1, 1];
  (b - a)/2*Sum[weights[[i, 2]] f[(a + b)/2 + (b - a)/2 weights[[i, 1]]],{i,1, n}]]

GaussLegendreQuadrature[0, Pi, n, g]

Is it correct?