In the present paper, we study a family of integrals of the form \[ R_{\mu}(k,\alpha,\nu)=\int^{\pi}_{0}\frac{\cos^{2\alpha - 1}(\theta /2)\sin^{2\nu -2\alpha -1}(\theta /2)d\theta}{(1-k^ 2\cos \theta)^{\mu +}} \] where \(0/\leq k<1\), \(Re(\nu)>Re(\alpha)>0\), \(Re(\mu)>-\). Special cases of \(R_{\mu}(k,\alpha,\nu)\) occur in radiation field problems. We obtain a series expansion of \(R_{\mu}(k,\alpha,\nu)\) and establish its relationship with Gauß hypergeometric function. Asymptotic expansions valid in the neighbourhood of \(k^ 2=1\), and some recurrence relations are given. Results obtained earlier by L. F. Epstein and the third author [J. Res. Natl. Bur. Stand., Sect. B 67, 1-17 (1963; Zbl 0114.064)], G. H. Weiss [ibid. 68, 1-2 (1964; Zbl 0122.067)] and the first author (see the paper reviewed above) follow as particular cases of our formulae established here. Some numerical values of \(R_{\mu}(k,\alpha,\nu)\) are computed.