We study the following three definite integrals, previously posed as open problems by another researcher: \ (I () = ₀^ x^-1/2 (1 + x^-) \, dx \), \ (Iₙ () = ₀^ 1x (x² + 4²) ⁿ\, dx \), and \ (I () =₀^ x^-3/2 (2/x) - f (2/x) \, 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 () turns out to be similar to Frullani integral, and we obtain two interesting formulas for this integral. These types ofintegrals have been used to establish logarithmic Hardy-Hilbert-type inequalities.
Irshad Ayoob (Fri,) studied this question.