It’s been said that we often remember our failures more than our successes. In this instance, the adage rings true, because I can still remember, clear as a bell, the one problem that I got wrong on my high school calculus final that I took 30 years ago. Here it is:
I tried every substitution under the sun, with no luck. I tried . However, would be equal to , and there was no extra in the integrand.
I believe I tried every crazy, unorthodox substitution possible given the time constraints of the exam: , , . Nothing worked.
We had learned trigonometric substitutions in my class, and so I also tried those. I started with , so that . This looked promising. However, , so the integral became . From there, I was stuck. (Now that I’m older, I know that the logical train actually goes in the reverse direction than what I attempted as a student.)
I wasn’t taught integration by parts in this first course in calculus, so I didn’t even know to try it. Had I known this technique, I probably would’ve broken through my conceptual barrier to finally get the right answer. (In other words, integration by parts will yield the correct answer, but it’s a lot of work!) But I didn’t know about it then, and so I get to tell the story now.
Exasperated, I turned in my exam when time was called, and I asked my teacher how this integral was supposed to be solved.
Easy, she told me: just square out the inside:
At the time, I was unbelievably annoyed at myself. Now, I love telling this anecdote to my students as I relate to their own frustrations as they practice the art of integration.