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.
You might consider the old warhorse
It’s very common in calculus texts to resort to the trick of multiplying and dividing by (secx+tanx), upon doing which the answer jumps right out with a bit of simplification. Any reasonable student, though, might complain about this “rabbit out of the hat approach,” asking, “How on earth could you expect me to come up with this idea?” All this approach does is impress the student with the author’s cleverness while at the same time making them feel stupid. Here’s an alternative approach that involves a different, and perhaps more accessible, kind of cleverness.
∫secx dx=∫1cosx dx=∫cosxcos2x dx=∫cosx1−sin2x dx=∫cosx(1(1−sinx)(1+sinx)) dx
Continue with partial fractions:
=∫cosx2(11−sinx+11+sinx) dx=12∫cosx1−sinx dx+12∫cosx1+sinx dx
and now two simple substitutions and a bit of algebra gives the result. Occasionally, after giving this version I’ll give the textbook version as an exercise, where it properly belongs.