  Szmytkowski derived a certain integral with Gegenbauer polynomials. A natural
generalization is to derive lookalike integrals with Jacobi polynomials. Six
methods are treated to derive the first integral. The first method should be
enough to prove the first integral, but by the other methods there arises
remarkable formula such as par example a zero-balanced F3 Appell function which
can be converted into a 2F1 hypergeometric function. Another three integrals
complete the paper.
