First of all,Unbounded integral never becomes bounded.Integral which you are referring as unbounded is actually comes under class of integrals known as Improper integrals.There are several techniques to evaluate them such as DUIS(Differentiation under integral sign),Residue Theorem,Laplace Transform,Fourier Transform(Improper Integrals involving trigonometric entities are also known as Fourier Integrals),Jordan Lemma.For more information you can refer this book :
Calculus with complex numbers :John B. Reade (Chapter 6 and 7)