The Laplace transform of the functions t ν ( 1 + t ) β , -1$"> Re ν > − 1 , is expressed in terms of Whittaker functions. This expression is exploited to evaluate infinite integrals involving products of Bessel functions, powers, exponentials, and Whittaker functions. Some special cases of the result are discussed. It is also demonstrated that the famous identity ∫ 0 ∞ sin   ( a x ) / x   d x = π / 2 is a special case of our main result.