An integral for the Prime

[Alberto Candel]

I have not been able to look into the giac source. I did look into the problem of evaluating some of these integrals of the form \(\int_0^1 R(x) (\log x)^m \, dx\), where \(R(x)\) is a power series in \(x\) satisfying certain conditions at \(x=1\). It does not seem to me that you need the residue formula, but only the value of integrals \(\int_0^1 x^n (\log x)^m \, dx = (- m)!/(n+1)^{m+1}\), which is easily obtained by parts. Integrals \(\int_1^\infty\) can be transformed to \(\int_0^1\) via the change \(x\to 1/x\), and integrals \(\int_0^\infty\) may be split like \(\int_0^1 + \int_1^\infty\). Many examples are given in Article 1071 of Edwards' Integral Calculus: the answer will depend on the series coefficients of \(R(x)\) and in many cases involves the Riemann and Dirichlet zeta functions (thus powers of \(\pi\) or Catalan's constant).

It doesn't seem to me that the Prime is using this approach for evaluation of those integrals above, as it can evaluate \(\int_0^\infty\) of some \(R(x) (\log x)^m\) but not \(\int_0^1\) of the same.