N. Nielsen and Ramanujan considered the integrals a_k = integral_1^2 (ln x)^k/(x - 1) d x. They found the values for k = 1 and 2. The general constants for k>3 were found by Levin and, much later, independently by V. Adamchik, a_p = p!ζ(p + 1) - (p(ln2)^(p + 1))/(p + 1) - p! sum_(k = 0)^(p - 1) (Li_(p + 1 - k)(1/2)(ln2)^k)/(k!), where ζ(z) is the Riemann zeta function and Li_n(x) is the polylogarithm.

polylogarithm | Riemann zeta function