Some estimates of the generalized beukers integral with techniques of partial fraction decomposition

In this paper we establish the generalized Beukers integral $I_{m}(a_{1},...,a_{n})$ with some methods of partial fraction decomposition. Thus one obtains an explicit expression of the generalized Beukers integral. Further, we estimate the rational denominator of $I$ and. In the second section of this paper, we provide some estimates of the upper and lower bound of the value $J_{3}$, which involves the generalized Beukers integral and is related to $\zeta(5)$.