Some integrals in Ramanujan's lost notebook

On page 199 in his lost notebook, Ramanujan recorded three unusual integral formulas. Two of his assertions are correct, and proofs are given here. His third claim is incorrect, but it is true in a limiting form. The integrals arise from a claim in Ramanujan's ordinary notebooks on the equality of certain integrals and series in which the integrand and the summands are themselves the same function.