Let \(Q^{(m)}f\) be the m-copy of the quadrature rule approximation Qf or \(Q^{(1)}f\) to the finite integral If having integrand f and interval [a,b]. For most rules Q and integrand functions f, \(Q^{(m)}f\) converges to If as m becomes infinite. For some Q and f, asymptotic expansions in m exist for the error functional \(Q^{(m)}f\)-If. The classical example is the Euler-Maclaurin expansion valid when Q is the trapezoidal rule and \(f\in C^{(p)}[a,b]\). This expansion, in inverse power of m, is the familiar one on which the classical version of Romberg integration is based. Several other expansions of this type have been reported since 1960. In particular, when f(x) has algebraic or logarithmic singularities at the endpoints a,b but is \(C^{(p)}(a,b)\), a significantly different expansion is known. In this paper, the theory is extended once more to cover functions like this, which may in addition have simple poles on the integration interval. In this case If is the Cauchy principal value integral. One result is that, when Q is the trapezoidal rule, \(f(x)=(x- c)^{-1}\phi (x)\), \(0=a