Logarithmic integrals with applications to BBP and Euler-type sums

Abstract

For real numbers p, q > 1 we consider the following family of integrals: 1 0 (x q−2 + 1) log (x mq + 1)x q + 1 dx and 1 0 (x pt−2 + 1) log (x t + 1)x pt + 1 dx.We evaluate these integrals for all m ∈ N, q = 2, 3, 4 and p = 2, 3 explicitly. They recover some previously known integrals. We also compute many integrals over the infinite interval [0, ∞). Applying these results we offer many new Euler-BBP-type sums.