Some Divergent Trigonometric Integrals

1. Introduction. Browsing through an integral table on a dull Sunday afternoon some time ago, I came across four divergent trigonometric inte-grals. (See (1) and (2) below.) I was intrigued as to how these divergent integrals ended up in a respectable table. Tracing their history, it turned out they were originally " evaluated " when some convergent integrals, (5) and (6), were differentiated under the integral sign with respect to a parameter, formally yielding (1) and (2). We will give a simple proof that these inte-grals diverge, look at their history in print and then make some final remarks about necessary and sufficient conditions for differentiating under the integral sign. We have no motive in defaming either the (shockingly famous) mathematician who made the original error, or the editors of the otherwise fine tables in which the integrals appear. We all make mistakes and we're not out to point the finger at anyone. (In this regard see the last two exercises of Chapter 2 in [30].) We will also see that Maple and Mathematica have considerable difficulties with these integrals.