Consequences of an infinite Fourier cosine transform-based Ramanujan integral

In this paper, we express a generalization of the Ramanujan integral I(alpha) with the analytical solutions, using the Laplace transform technique and some algebraic relation or a Pochhammer symbol. Moreover, we evaluate some consequences of a generalized definite integral phi * (upsilon, beta , a). The well known special cases appeared whose solutions are possible by Cauchys residue theorem, and many known applications of the integral I(a, beta , upsilon) are discussed by Leibnitz's rule for differentiation under the sign of integration. Further, one closed-form evaluation of the infinite series of the F-1(0) (<middle dot>) function is deduced. In addition, we establish some integral expressions in terms of the Euler numbers, which are not available in the Gradshteyn and Ryzhik