I must find the simpson's rule by integrating the interpolant polynomial of . The simpson's rule states that .

I have some data : , and .

Therefore by a theorem, there exist an unique polynomial of degree or equal to 2 passing by .

I won't write all, but finding the polynomial via Lagrange's form is not so hard. I found

Developing the expression, I don't come that close from the simpson's rule. I have a factor 4 in front of but it is still multiplied by a complicated expression that I'm not able to get rid of. And in front of f(a) and f(b) I don't have any common factor, but an expression with and so on. Still can't get rid of this. Can anyone help me finding the simpson's rule?