Theorem (Gauss-Legendre Quadrature).  An approximation to the integral  

        

is obtained by sampling    at the    unequally spaced abscissas   , where the corresponding weights are   .  
The abscissa's and weights for Gauss-Legendre quadrature are often expressed in decimal form.

n=2 Rule      
where      
                

n=3 Rule       
where      
                
                

n=4 Rule      
where      
                
                
                

n=5 Rule      
where      
                
                
                
                

Remark. For ease of reading the above list of rules has used the notation and instead of   and , respectively.

Truth.

    The abscissas and weights for the rules n=2,3,4,5 can be expressed with radicals.  

n=2 Rule    

n=3 Rule    

n=4 Rule    

n=5 Rule      

The higher rules for n>5 only use decimal values for the abscissas and weights.

n=6 Rule    
where      
                
                
                
                
                

n=7 Rule    
where      
                
                
                
                
                
                

n=8 Rule    
where      
                
                
                
                
                
                
                

(c) John H. Mathews 2004