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I am very interested in evaluating difficult definite integrals without elementary antiderivatives by manipulating the integral somehow (e.g. contour integration, interchanging order of integration/summation, differentiation under the integral sign, etc.), especially if they have elegant solutions. However, I simply cannot seem to find a good book that covers many ways to evaluate these. The single book that I know that covers some good techniques is the Schaum's Advanced Calculus.
What is another good book that explains methods and techniques of integration of these fun integrals?

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Integrating tricky things in an elegant way is a surprising and fun art. Yeah, I would like to hear about the Gourmet Guide to Integration. – ncmathsadist Jul 27 '12 at 23:47
Are you asking about definite or indefinite integration? And about "tricks" useful for specially-devised problems (e.g. competitions) or about general techniques for integral that occur in the wild? – Bill Dubuque Jul 28 '12 at 0:23
@BillDubuque I was referring to definite integrals (edited accordingly). I am fine with both specially-devised problems and general teqniques, or both. – Argon Jul 28 '12 at 1:08
up vote 9 down vote accepted

I've only skimmed it, but Irresistible Integrals by George Boros and Victor H. Moll seems worth a look.

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My favorite is a book of 1992 from Daniel Zwillinger : "Handbook of Integration" it is a "compilation of the most important methods" in 360 pages. Recommanded!

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(+1) for introducing me to this interesting text. – user26872 Jul 29 '12 at 0:57
@oen: thanks oen! Of course I have the other three (now) excellent books too (how could I miss a Feynman book anyway? even... if completed with complex integration! :-) You got my vote yesterday of course!), – Raymond Manzoni Jul 29 '12 at 9:48

As a first approximation, Whittaker and Watson's "... Modern Analysis" shows how to use the Gamma function, and other classical "special functions" (that one would not hear about in traditional calculus classes,...) to address such issues.

If Whittaker and Watson were Baroque, the standard reference "Gradshteyn and Ryzhik" (sp?) would be the Rococco. They do give serious references for all their formulas... and the whole thing is amazing...

A curiously small, tractable, readable, inexpensive, available source is Lebedev's book on special functions.

Vilenkin's AMS translation book about special functions "versus" representations is chock-full-of stuff.

(And, perhaps unsurprisingly, given that I'm responding, various notes of mine on my web site talk about "how to evaluate integrals". At there is a straightforward discussion of certain not-really-elementary integrals that arise in number theory/autormophia... and might be of interest anyway. Other course notes on my various web pages address other aspects of similar things in that context... which may not be your main interest...)

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Mathews and Walker's Mathematical methods of physics has some good tricks for integrals, among many other things. They give numerous natural examples and problems that use the methods of contour integration, differentiating under the integral sign, symmetry arguments, Feynman parameters, approximating integrals, etc. They also treat many of the special functions. The text grew out of lectures by Richard Feynman.

As an example, here's an integral over solid angle treated in that text $$\int\frac{d\Omega}{(1+{\bf a}\cdot {\bf \hat r})(1+{\bf b}\cdot {\bf \hat r})},$$ where $\int d\Omega (\ldots) = \int_0^{2\pi}d\phi \int_{-1}^1 d(\cos\theta)(\ldots)$. This can actually be written in terms of elementary functions, but to do so is not so easy!

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