The use of a connection between integral transforms for the computation of integrals

A theorem is proved which establishes a connection between four well-known integral transforms: Laplace, Kantorovich-Lebedev, Mehler-Fok, and the K-transform of Meier. This theorem is used to calculate a number of integrals containing the Legendre function β_1/2+iτ(X), and also the MacDonald function K_iτ(X). Certain other integrals can be calculated by using the indicated method.