This analysis evaluates certain infinite integrals continuing products of Bessel functions of integer order, an exponential and a power. The integrals considered here have been previously evaluated in the literature in two different forms. In one instance they have been written in terms of complete elliptic integrals of the first, second, and third kind. Some of these integrals have also been evaluated in terms of a Legendre function of the second kind and a complete elliptic integral of the third kind. A recent result in elasticity obtained by the authors has led to a new form for the evaluations of these integrals. The integrals are still evaluated in terms of complete elliptic integrals; however, a new modulus (and parameter for the complete elliptic integral of the third kind) is used. The new form used for the complete elliptic integral of the third kind allows the integral evaluations to be written in a more convenient form than previously given. The new form for the complete elliptic integral of the third kind is also utilized in the evaluations using the Legendre function of the second kind. The new forms to the integral evaluations derived presently are correlated with existing results in the literature.