On the Evaluation of Certain Unsolved Definite Integrals

We study the following three definite integrals, previously posed as open problems by another researcher: \( I(α) = ∫_0∞ x-1/2 ln(1 + x-α)\,dx \), \( I_n(α) = ∫_0∞ {1}{√x(x^2 + 4α^2)^n}\,dx \), and \( I(β)=∫_0∞ x-3/2 [f(2β/x) - f(2/x)]\,dx \). 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(β) 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.