Evaluating $I(n) = \int^{\infty}_{0} \frac{\ln(x)}{x^n(1+x)}\, dx$ for real $n$ I am not sure how to handle the additional parameter $n$. I first need to find out for which real values of $n$ will the integral converge. Based on intuition and checking with mathematica, I believe it will converge only for $0 < n < 1$, although I am not sure and I would like to see exactly how to determine those values. For this integral, I would normally try using a branch cut, but again I am not sure how to apply it given my additional parameter $n$.
$$I(n) = \int^{\infty}_{0} \dfrac{\ln(x)}{x^n(1+x)}\, dx$$

Firstly, to determine the range of $n$ for which the integral converges, just look at the singularity at $0.$ To integrate, and put the pesky $n$ where it is easier to deal with, make the substitution $u=\log x.$ This will transform your integral to $\int_{-\infty}^{\infty} u \exp(-(n-1) u)/(1+\exp u) du,$ which should be easy by standard contour integration methods. To check your computation, the answer is $\pi^2 \cot(n \pi) \csc(n \pi),$ provided $0<n<1.$
