Yesterday I asked for the derivation of the Integral representation of the Digamma-Function: https://math.stackexchange.com/questions/3113119/integral-representation-of-digamma-function

Thanks again @Jair Taylor for the great and detailed explanation.

Today, however, I stumbled across the Integral representation of the Polygamma-Function on Wikipedia:

$\psi^{(n)}(x)=(-1)^{n+1}\int _{0}^{\infty }\left({\frac {t^{n}e^{-xt}}{1-e^{-t}}}\right)\,dt$

For $n=0$ this should represent an Integral representation for the Digamma-Function. So if I insert $n=0$ into the equation I end up with:

$\psi^{(0)}(x)=(-1)\int _{0}^{\infty }\left({\frac {e^{-xt}}{1-e^{-t}}}\right)\,dt$

This, however, is different from the Integral representation of the Digamma-Function I ended up with yesterday:

$\psi (x)=\int _{0}^{\infty }\left({\frac {e^{-t}}{t}}-{\frac {e^{-xt}}{1-e^{-t}}}\right)\,dt$

So how is it possible that the two Integral representation differ by the factor $\frac {e^{-t}}{t}$ within the Integral and still both are correct?

Please apologize if the solution is kind of obvious, I thought a lot about it and still don't see it.

Thank you so much for your help :)