Beukers [3] used some double integrals to give an elegant proof to Apéry's result, which states that $zeta(3)$ is irrational. In this paper, based on his methods, we generalize Beukers' integrals (although we do not prove the irrationality of $zeta(2n+1)$ for positive integer $n$). The evaluation of these integrals is achieved by  using an
expansion of an infinite geometric series and differentiating under the integral sign.

Keywords: Beukers&rsquo, integers, irrationality of numbers, Riemann zeta function