I'm nearing the end of my studies in Calculus BC (or I and II). And, every now and then, I encounter somewhere a really clever trick to an integral that wasn't formally taught to me but was perfectly comprehensible for a student studying single-variable calculus (e.g. Feynman Method, Euler Substitution, Differentiating inside the Integral Sign, Weierstrass Substitution, etc.). Now that I am nearing my AP test, I feel like I'm also in need of a bit more practice with integrals. I am wondering if there is a bag of (potentially esoteric) tricks somewhere with accompanying exercises and full solutions that deal with limits, integrals, weird trig and hyperbolic integrals, differential equations, series convergence, etc. (within the scope of single-variable calculus). I'd like it to be relatively challenging but also rewarding for the tricks that it introduces. I want to feel like I really mastered the subject at the end of things, and have the skills to be able to potentially participate in a contest of some sort to add to my college application. Thank you in advance for your inputs.
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Are old practice AP tests available? – user2661923 Jun 14 '24 at 17:42
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They are, but I'm looking for something a bit more fun which involves some fun tricks that usually aren't taught (like Euler Substitution). – procommania Jun 14 '24 at 17:50
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See my answer to Sources for reading about integration techniques. However, you should keep in mind that the AP test is extremely basic regarding to integration techniques needed. – Dave L. Renfro Jun 14 '24 at 17:54
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I'm not in the last stretches of test preparation yet. I do still have some time until the May test. I just want to learn some of the more exotic tricks of calculus and get some practice. – procommania Jun 14 '24 at 17:55
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The Nahin one seems very enticing. I'll definitely need some time to peruse these suggestions. Thank you! – procommania Jun 14 '24 at 18:05
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There's a user named Quanto on this site who normally solves integrals using the methods you have described, and I find their solutions interesting. My friend also has a YouTube channel about math tricks in this link here. Other than them, just look through the tabs that mention something integration or series related. – Accelerator Jun 15 '24 at 01:02
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1I did actually watch his videos before. They felt very enlightening. I've just discovered his integration bee playlists. I'll try to see if I can follow. – procommania Jun 15 '24 at 01:18