Does anyone have a good reference for a book that already assumes knowledge of calculus/analysis and whose main focus is computing more difficult integrals? I'm looking for something which will have a lot of worked examples for differentiation under the integral, tricky substitutions, unusual contours, etc.

Most of what I've found in terms of references simply compile a long list of integrals, many of which are more tedious than insightful. My goal is to be able to get better at computing integrals that can be done by hand in a reasonable amount of time, but may require some ingenuity (like in math competitions, for example). So any discussion of heuristics and problem solving techniques in integration is a plus.


You might be interested in The Handbook of Integration by Zwillinger. It appears to be the standard reference on integration methods for scientists and engineers.

The downside is that it probably doesn't contain the "tricky" techniques you are looking for. Hopefully someone can find a more math contest-oriented book for you.

In the meantime, you might try looking at some answers this site. We have many talented integration experts around who are frequently posting and answering extremely challenging integrals and sums (like this). Here's an incomplete list (go to the 'answers' tab on their profile):

  • sos440 (He also maintains a blog, which has many challenging integrals worked out in complete detail.)

  • Mhenni Benghorbal (His answers are especially good for learning Mellin transform techniques.)

  • Ron Gordon (He uses mostly complex analysis. Also check his 'greatest hits' page, available in his profile.)

You may also want to check the questions of 'Chris's sis', the asker of the example question I linked above, though many of her questions are not integrals.

Bennett Gardiner has mentioned this useful site in a comment. On this site, there is a link to a useful guide.

  • $\begingroup$ Also, these guys or at least people like them all hang out on a website that I can't seem to find now, called Integration and Series, or something like that, and just pose and answer tricky integrals and series to/for each other. It's really awesome and I'm really annoyed that I can't remember the specific URL. Anyone know what I'm talking about? $\endgroup$ – Bennett Gardiner Oct 18 '13 at 6:29
  • $\begingroup$ Also, wanted to reaffirm Ron's mastery of contour integrals - and just now seeing a few examples of Mhenni's Mellin transforms - amazing! So clean and elegant. $\endgroup$ – Bennett Gardiner Oct 18 '13 at 6:44
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    $\begingroup$ @BennettGardiner I found the site and added it to my answer. Thanks for the suggestion. $\endgroup$ – Potato Oct 18 '13 at 6:49

I've considered buying this book many times -


perhaps you will, and let me know how it is!

Edit - although contour integrals are avoided, real methods only. Reading Ron's "best of" and using any good complex variables book (Ablowitz and Fokas recommended) should give plenty of practise at that.

Actually the wikipedia page http://en.wikipedia.org/wiki/Methods_of_contour_integration has heaps of examples.


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