I am teaching Calc 1 right now and I want to give my students more interesting examples of integrals. By interesting, I mean ones that are challenging, not as straightforward (though not extremely challenging like Putnam problems or anything). For example, they have to do a $u$-substitution, but what to pick for $u$ isn't as easy to figure out as it is usually. Or, several options for $u$ work so maybe they can pick one that works but they learn that there's not just one way to do everything.
So far we have covered trig functions, logarithmic functions, and exponential functions, but not inverse trig functions (though we will get to this soon so those would be fine too). We have covered $u$-substitution. Thinks like integration by parts, trig substitution, and partial fractions and all that are covered in Calc 2 where I teach. So, I really don't care much about those right now. I welcome integrals over those topics as answers, as they may be useful to others looking at this question, but I am hoping for integrals that are of interest to my students this semester.