I'm mixed when I accrossed the formula $(01)$ is presented in this paper (formula (6.1)) in the last page ) and formula $(02)$ is presented in this paper (page 2, formula 3) as shown below , These formula were montioned in the two papers that they are proved By Tchebeychev for $Re(s) >1$ , but they are not similar , I have tried to show that they are equivalent but i don't succeed , i don't know where is the problem ? Are they the same ? and if they are how do i show that they are equivalent ?
$$\zeta(s)+1-\frac{1}{s-1}=\frac{1}{\Gamma(s)}\int_0^\infty \displaystyle(\frac{1}{-1+e^x}-\frac{1}{x})e^{-x}dt \tag{01}$$
$$\zeta(s)-1-\frac{1}{s-1}=\frac{1}{\Gamma(s)}\int_0^\infty (\frac{1}{-1+e^x}-\frac{1}{x})e^{-x}x^{s-1}dt \tag{02}$$