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It's coming up to Christmas so I can ask to have all the books I can't afford from begrudging relatives! I'm really interested (mainly from looking at some of the answers cleo and other fantastic users!) in being able to approach integrals from a variety of different ways and learning how to tackle non-elementary integrals.

I've gone over a lot of the standard techniques in my undergrad and this is just for a hobby, so don't want anything too 'heavy', just great explanations and a lot of questions to tackle. So far I've found Irresistible Integrals: Symbolics, Analysis and Experiments in the Evaluation of Integrals. Many thanks.

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    $\begingroup$ Hmmm. This does not quite answer your question, since it does not really have explicit techniques or questions to tackle. However, someone has taken the effort to prove many of the formulas in the famous Gradshteyn and Ryzhik. I imagine one could pick up many interesting techniques from reading these proofs. Here it is, if it happens to be of interest to you: 129.81.170.14/~vhm/Table.html $\endgroup$ – Alex Wertheim Nov 28 '14 at 0:35
  • $\begingroup$ @AWertheim Thanks - what a database! $\endgroup$ – Mike Miller Nov 28 '14 at 9:30
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    $\begingroup$ Maybe a bit on the "heavy" side, but you can find some book suggestions here: matheducators.stackexchange.com/questions/3950/… $\endgroup$ – Hans Lundmark Nov 29 '14 at 10:38
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    $\begingroup$ @AWertheim: "Someone" is Victor Moll, one of the author of the book "Irresistible Integrals" mentioned in the question. $\endgroup$ – Hans Lundmark Nov 29 '14 at 10:41
  • $\begingroup$ @HansLundmark Thank you - I'll have a Google of some of the suggestions. $\endgroup$ – Mike Miller Nov 29 '14 at 14:09
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I wouldn't consider my book great or elementary but have a look at it

Advanced integration techniques

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  1. This question has been asked and answered elsewhere, so check for similar threads here. [Not meant to police the question, just FYI more answers there.]

  2. Joseph Edwards Integral Calculus Treatise (old enough to be out of copyright and free pdfs available) has 1000+ pages of integration techniques and problems (many from the Cambridge Tripos).

  3. Check out Zwillinger handbook or other handbooks of Integrals. While not as fun as a problem list, you can learn from these books also. Especially the ones that do more than just list formulas (like CRC tables) but give some derivation or other comments.

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  • $\begingroup$ I appreciate point 1 but it is over 3 years since I posted this :P $\endgroup$ – Mike Miller Jan 25 '18 at 21:27
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If after three years your interest in evaluating definite integrals has not waned, you may consider consulting the following texts:

  1. Improper Riemann integrals by Ioannis M. Roussos (CRC Press, 2014).

  2. Solved problems: Gamma and beta functions, Legendre polynomials, Bessel functions by Orin J. Farrel and Bertram Ross (MacMillian, 1963).

  3. Integration for engineers and scientists by W. Squire (Elsevier, 1970).

  4. Integral evaluations using the gamma and beta functions and elliptical integrals in engineering: A self-study approach by C. C. Maican (International Press, 2005).

  5. An introduction to sequences, series, and improper integrals by O. E. Stanaitis (Holden-Day, Inc. 1967).

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