On the Evaluation of Certain Unsolved Definite Integrals

Abstract

We study the following three definite integrals, previously posed as open problems by another researcher: \( I(\alpha) = \int_0^{\infty} x^{-1/2} \ln(1 + x^{-\alpha})\,dx \), \( I_n(\alpha) = \int_0^{\infty} \frac{1}{\sqrt{x}(x^2 + 4\alpha^2)^n}\,dx \), and \( I(\beta)=\int_0^{\infty} x^{-3/2} \big[f(2\beta/x) - f(2/x)\big]\,dx \). We establish sufficient conditions for the convergence of these integrals and evaluate them in
closed form using special functions. In particular, the third integral $I(\beta)$ turns out to be
similar to Frullani integral, and we obtain two interesting formulas for this integral. These types of
integrals have been used to establish logarithmic Hardy-Hilbert-type inequalities.