On the evaluation of a definite integral involving nested square root functions

There are many and varied methods one can employ when seeking the evaluation of definite integrals in closed form. In many circumstances the method employed reflects some intrinsic property of the integrand in question. For example, certain symmetry considerations may be exploited as in the case of determining the convergent improper integral of e-x2 over R. In like manner, we shall in this paper take advantage of a structural feature of a class of nested square root functions, in order that we may evaluate their corresponding definite integrals which on first acquaintance appear rather intractable.