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This paper provides new results for a family of incomplete Lipschitz-Hankel integrals (ILHIs) which also lead to the evaluation of certain integrals involving the generalized Marcum Q function. These mathematical results are then applied to analyze the bit error rate (BER) of adaptive modulation over multiple-input-multiple-output (MIMO) fading channels under imperfect channel state information (CSI). A novel exact closed-form expression for the average BER of adaptive modulation under MIMO transmit beamforming with maximal ratio combining, assuming prediction errors at the receiver for the adaptation CSI required by the transmitter, is obtained. The benefit of this result with respect to previous analysis is threefold. First, the expression is an exact closed form. Second, it is applicable to any antenna configuration, and third, it allows a design improvement of the cutoff SNR thresholds, which leads to better system performance in terms of average spectral efficiency at no extra cost.