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 $\beta_{\frac12+i\tau}(x)$, and also the MacDonald function $K_{i\tau}$. Certain other integrals can be calculated by using the indicated method.