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I want to learn to solve integrals of some type, probably definite integrals with results involving various constants such as Catalan's, Euler-Mascheroni,Golden-ratio etc. and involving various functions like hyperbolic, inverse hyperbolic, hypergeometric, zeta etc. but can't find any resources on them. I found various sites with integration techniques but most of them list basic integration formulas.I am looking for something outside the elementary functions.Wikipedia and Wolfram are sudden spikes of unexplained complex results.

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Maybe this can be of some assistance, Kokeboken.pdf if you dissregard the alien language. – N3buchadnezzar Sep 3 '14 at 18:14
up vote 6 down vote accepted

`Irresistible Integrals' by Boros and Moll should fit your bill. I remember that integrals involving Apery's constant, Bernoulli numbers, Euler's constant and the like feature in it.

Moreover, it looks at integrals from a wider variety of sources (e.g. number theory) than traditional `Tables of integrals' like Gradshteyn and Ryzhik.

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looking at contents make me much happy. – ADG Sep 3 '14 at 17:17
Incidentally, Amazon also suggested this unreleased title. Appealing. – Mark Fantini Sep 3 '14 at 18:56
do you have a pdf of Irr.Int.? – ADG Sep 5 '14 at 18:02

Wikipedia and Wolfram are sudden spikes of unexplained complex results.

Odd that you've mentioned the word “complex”. Indeed, complex integration techniques, taught in college, whose “crown jewel” is the residue theorem, are the key to obtaining many of these results. Other methods consist in expanding the integrand into its Taylor or binomial series, then switching the order of summation and integration. Yet other approaches rely on various transforms, such as Fourier, Laplace, Mellin, or Z. Then there is the one championed by Feynman, and discovered by Leibniz, namely differentiation under the integral sign. Also, a similar question has been asked here.

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i meant complex in sense of a linguistic synonym – ADG Sep 3 '14 at 17:41
@Aditya: Yes, I know what you meant. :-) – Lucian Sep 3 '14 at 17:42

If you are looking for free source, you may refer to the collection of proof of the integrals in Gradshteyn and Ryzhik. Just click the green link $\color{green}{\text{Proof}}$ to download the file.

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