Summary: We find that the solution of the polar angular differential equation can be written as the universal associated Legendre polynomials. Its generating function is applied to obtain an analytical result for a class of interesting integrals involving complicated argument, that is, \(\int_{- 1}^1 P_{l^\prime}^{m^\prime} ((x t - 1) / \sqrt{1 + t^2 - 2 x t}) (P_{k^\prime}^{m^\prime}(x) /(1 + t^2 - 2 t x)^{(l^\prime + 1) / 2}) d x\), where \(t \in(0,1)\). The present method can in principle be generalizable to the integrals involving other special functions. As an illustration we also study a typical Bessel integral with a complicated argument \(\int_0^{\infty} (J_n(\alpha \sqrt{x^2 + z^2}) / \sqrt{(x^2 + z^2)^n}) x^{2 m + 1} d x\).