# What is the terminology for "self-referral" integrals in calculus?

In the topic of integration and anti-derivatives in Calculus we come across cases where the attempt at integration by parts brings us back to the original integral, the most basic example being $$\int e^{a x} \cos(b x) \,dx$$. Is there a standard name for this type of integrals? I have been calling them "self-referral", but the only usage of this expression that I can find is in health insurance! Have you seen a name in standard textbooks? I have seen "circular integrals" online. What do you call them? Does anyone call it "self-referential"? (Stewart does not have a name for it. Self-referential is an expression in logic. Circular integral might be used for contour integrals.)

• I have never heard of a term for such a thing, nor do I think that they are important enough to bother with a specific name. However, I would avoid the term "circular integral", as I would understand such an integral to either be an integral over a circle, or an integral related to some specific subset of elliptic integrals (not that I know what specific set this would be---that would just be my assumption about the term). Jan 19 at 18:33
• "self-referential" seems fine. I love how those turn into algebra problems, which is part of what I emphasize. Jan 19 at 19:04
• Giving it a name would give the (wrong) impression that it's a feature of the integral itself, whereas it's only a feature of this particular technique we're using to derive a closed form expression for this integral. PS. Justin Skycak's answer provides a name which is free of this issue. Jan 19 at 20:11
• Call me a simpleton, but my programming background tells me that you arrive at an equation involving recursion. Jan 20 at 11:08
• On Mathematics Stack Exchange, it was referred to as "Integration by parts with déjà vu". I found this terminology amusing enough that I use it myself whenever I tutor school students.
– Joe
Jan 21 at 0:23