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I finished calculus II and want to learn more about techniques of indefinite integrals . Which book should I read ?

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    $\begingroup$ I have read the book : "Mathematical Analysis II, Integral Calculus" By Sever Angel Popescu in my course advanced calculus II. I think this will help you a lot. A pdf is available on: civile.utcb.ro/cmat/cursrt/ma2e.pdf $\endgroup$ – Fibonacci Feb 28 '18 at 16:04
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    $\begingroup$ I would recommend a book about measure theory. "Measure and Integration Theory" by Heinz Bauer is a good way to start this topic. $\endgroup$ – Ben373 Feb 28 '18 at 16:30
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    $\begingroup$ Based on the two comments already given, you should probably clarify what you mean by "techniques of integration". Do you mean explicit algebraic methods (integration by parts, trig substitutions, etc.) or do you mean theoretical techniques such as Darboux sums, Riemann sums, etc.? $\endgroup$ – Dave L. Renfro Feb 28 '18 at 18:37
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    $\begingroup$ Perhaps you're interested in this sort of thing? smile.amazon.com/… $\endgroup$ – Adrian Keister Feb 28 '18 at 20:04
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If you are interested in learning the various techniques of integration (for Riemann integrals) at a level comparable to Calculus I, II, and a bit beyond, together with practicing these techniques on many examples, you may consider:

  1. Problems in Mathematical Analysis edited by Boris Demidovich. It contains many exercises using the various techniques of integration to practice on. A few of the techniques Demidovich gives are not usually found elsewhere.

  2. Joseph Edwards’ A Treatise on the Integral Calculus (Volume 1) is a particularly valuable source for many interesting integrals.

  3. Michael Spivak’s Calculus contains many interesting questions that use integration, like the proof that $\pi$ is irrational for example.

  4. How to Integrate It: A Practical Guide to Solving Elementary Integrals by Seán M. Stewart has individual chapters devoted to a particular technique of integration, with each being accompanied by a wealth of end-of-chapter exercises.

If, however, you are after techniques that can be applied to (Riemann) improper integrals you may consider:

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

  2. Inside Interesting Integrals, A Collection of Sneaky Tricks, Sly Substitutions, and Numerous Other Stupendously Clever, Awesomely Wicked, and Devilishly Seductive Maneuvers for Computing Nearly 200 Perplexing Definite Integrals From Physics, Engineering, and Mathematics by Paul J. Nahin (Springer 2014).

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

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

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

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