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There is an old $(\text{circa } 1930)$ and interesting book in calculus: Edwards' Treatise on Integral Calculus. This book has a very complete list of cases of integrals, for example, these monstrosities:

  • Cases of $\displaystyle \int\frac{x^n dx}{\prod_{1}^{n}(x^2+a_r^2)}$.

  • Hermite integration of $\displaystyle \int \frac{f(\sin \theta, \cos \theta)}{\prod_{1}^{n}\sin(\theta - a_n)}d\theta$.

Nowadays, calculus books have way less integrals than the aforementioned book. I wonder why is that. From the top of my head, I have some possible reasons and a refutation for some:

  • Those integrals could be very complicated and as we could easily calculate with a computer, they are not worth teaching anymore? Refutation: If the argument follows, It's actually easy to also compute most of the integrals in a modern calculus text using the computer and then, if we abandon those two examples, we could/should(?) abandon all of them.

  • Those integrals were important for some field of study in the past but this field of study is not relevant anymore?

  • We can calculate those by decomposing them into the few integrals we study today with modern calculus textbooks?

I have little knowledge in both calculus and the history of calculus to answer this question or to formulate better possible reasons. I am open to answers and also articles/books about the history of calculus textbooks (supposing they exist).

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  • $\begingroup$ Your premise is interesting but I would like to see more support for its truth. Maybe you are comparing the definitive guide to integral calculus with random introductory textbooks? $\endgroup$ – hkBst May 24 '16 at 8:06
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    $\begingroup$ In a calculus textbook the different examples are there to teach the reader the essence of integration. Some integrals are too important not to mention. But other than that who needs list books anymore? When you have everything at your disposal with one simple google search. $\endgroup$ – Hamed May 24 '16 at 8:06

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