Summary: Let \(TA(f)=\int^ b_ a{1\over 2}(P^ n_ a(x)+P^ n_ b(x))dx\) and let \(TM(f)=\int^ b_ aP^{n+1}_{(a+b)/2}(x)dx\), where \(P^ n_ c\) denotes the Taylor polynomial to \(f\) at \(c\) of order \(n\), where \(n\) is even. \(TA\) and \(TM\) are reach generalizations of the trapezoidal rule and the midpoint rule, respectively, and are each exact for all polynomials of degree \(\leq n+1\). Let \(L(f)=\alpha TM(f)+(1-\alpha)TA(f)\), where \(\alpha={2^{n+1}(n+1)\over 2^{n+1}(n+1)+1}\), we obtain a numerical integration rule \(L\) which is exact for all polynomials of degree \(\leq n+3\). The case \(n=0\) is just the classical Simpson's rule. We analyze in some detail the case \(n=2\), where our formulae appear to be new. By replacing \(P^{n+1}_{(a+b)/2}(x)\) by the Hermite cubic interpolant at \(a\) and \(b\), we obtain some known formulae by a different approach. Finally we discuss some nonlinear numerical integration rules obtained by taking piecewise polynomials of odd degree, each piece being the Taylor polynomial of \(f\) at \(a\) and \(b\), respectively. Of course all of our formulae can be compounded over subintervals of \([a,b]\).