\begin{align*} \int_{0}^{\infty}\left(x^{-1}e^{-x}-\frac{e^{-nx}}{1-e^{-x}}\right)dx & =\left[\log\left(x\right)e^{-x}\right]_{0}^{\infty}+\int_{0}^{\infty}\left(\log\left(x\right)e^{-x}-e^{-nx}\sum_{k=0}^{\infty}e^{-kx}\right)dx\\ & =-\lim_{x\rightarrow0}\log\left(x\right)+\left[\frac{d}{dt}\int_{0}^{\infty}x^{t}e^{-x}dx\right]_{t=0}-\int_{0}^{\infty}\sum_{k=0}^{\infty}e^{-\left(k+n\right)x}dx\\ & =-\lim_{x\rightarrow1}\log\left(1-x\right)+\left[\frac{d}{dt}\Gamma\left(t+1\right)\right]_{t=0}+\sum_{k=0}^{\infty}\frac{1}{k+n}\left[e^{-\left(k+n\right)x}\right]_{0}^{\infty}\\ & =\lim_{x\rightarrow1}\sum_{k=1}^{\infty}\frac{x^{k}}{k}+\left[\Gamma\left(t+1\right)\frac{d}{dt}\log\Gamma\left(t+1\right)\right]_{t=0}-\sum_{k=0}^{\infty}\frac{1}{k+n}\\ & =\sum_{k=1}^{\infty}\frac{1}{k}+\left[\Gamma\left(t+1\right)\psi\left(t+1\right)\right]_{t=0}-\sum_{k=1}^{\infty}\frac{1}{k+n-1}\\ & =\sum_{k=1}^{\infty}\frac{1}{k}+\Gamma\left(1\right)\psi\left(1\right)+H_{n-1}-\sum_{k=1}^{\infty}\frac{1}{k}\\ & =\Gamma\left(1\right)\psi\left(1\right)+H_{n-1}\\ & =H_{n-1}-\gamma \end{align*}