On integrals arising from a Lie group contraction problem

In a paper by the reviewer and S. K. Gupta [Monatsh. Math. 128 (1999), no. 3, 237–253; MR1719423], estimates for certain integrals were needed, particularly in an attempt to extend the work to other compact groups. These integrals are of the form
Ia,θ(n)=∫nθ−nθ|x|∣∣∣∣∣sinx2n2+a2−−−−−−√x2n2+a2∣∣∣∣∣⎛⎝⎜1−x2n2sin2x2n2+a2−−−−−−√x2n2+a2⎞⎠⎟n2dx.

In the paper under review the authors calculate the actual values of limn→∞Ia,θ(n). The limits depend upon the relationship between θ and a.