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We start with the computation of the Nuttall Q-function using Gauss quadrature rules. Since the integrand of the Nuttall Q-function is characterized by an exponential kernel function, the semi-infinite Gauss-Hermite quadrature is employed to efficiently calculate the tail probability involved. As such, with the help of the Gauss-Legendre quadrature, the Nuttall Q-function with arbitrary real orders can be accurately approximated in closed form. This methodology is then extended to provide closed-form approximations for other related semi-infinite integrals encountered in performance analysis of energy detection in fading channels.