[vcarl]

I remember a similar problem in Calc 2. I forget the specifics now, but I think it was an integral of some combination of sin/cos that ended up being circular. You had to recognize an opportunity to swap one of the steps for an equivalent, which would lead you to the final solution.

[mattb314]

Probably the second example here [1] for those curious (I think the integral of sin(x)*e^x dx is the only place I've seen this used, would love to know if there are other examples).

[1] https://en.m.wikipedia.org/wiki/Integration_by_parts#Tabular...

[octonion]

This becomes much more transparent if you realize you're integrating Im(e^x * e^{ix}). And it's no longer a trick but a technique.

[vcarl]

It would be great if that were how it was taught, but when I took the class it was taught as a trick. No theory behind it, just the prof on the board saying, "But look! :swap: And now you can integrate it."

[AstralStorm]

It becomes a technique once you realise it is a specific case of change of bases. Even just getting to basic theorem of algebra does not provide this insight.