Evaluating this indefinite integral

Question

This is something I came up with off the top of my head, not a homework problem, so it might not have a simple solution. $$\int_1^\infty ln(\frac{t-x}{t})dt$$ Where $x$ is a positive real number. I was able to find the antiderivative: $$t\ln(\frac{t-x}{t})-x\ln(t-x)$$ Which was pretty straightforward but got caught up trying to evaluate it at infinity. It seems like the first term should be either finite or zero, but the second term is definitely infinite. Yet Wolfram says it converges, so what is the proper way of evaluating it?

Answer 1

\begin{align*} t\ln\left(\frac{t-x}{t}\right)-x\ln(t-x)&=\ln\left(\frac{(t-x)^t}{t^t}\right)-\ln\left((t-x)^x\right)\\ &=\ln\left(\frac{(t-x)^{(t-x)}}{t^t}\right)\\ \end{align*}

For $x>0$, as $t\rightarrow\infty$, this diverges to $-\infty$. For $x<0$, as $t\rightarrow-\infty$, this diverges to $\infty$.