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I am a third year undergraduate mathematics student.

I learned some basic techniques for simplifying sums in high school algebra, but I have encountered some of the more interesting techniques in my combinatorics classes and math contests. Many of my favorite techniques involve showing some sort of bijection between things.

However, I feel that I have learned almost no new cool integration technique since I took the AP Calculus exam in high school. The first combinatorics book I remember reading had a large chunk devoted to interesting techniques for evaluating summations, preferably with bijective techniques. I have yet to encounter a satisfying analog for integrals.

There are two main things I have had difficulty finding out much about:

  1. What "subject" (perhaps a course I can take, or a book I can look up) might I look into for finding a plethora of interesting techniques for calculating integrals (e.g. for summations I might take a course in combinatorics or read "Concrete Mathematics" by Knuth et al)?

  2. I am particularly interested in analogs for "bijective proofs" for integrals. Perhaps there are techniques that look for geometric interpretation of integrals that makes this possible? I often love "bijective proofs" because there is often almost no error-prone calculi involved. In fact, I often colloquially define "bijective proofs" this way--as any method of proof in which the solution becomes obvious from interpreting the problem in more than one way.

I don't know how useful it would be to calculate interesting (definite or indefinite) integrals, but I feel like it would be a fun endeavor to look into, and as a start I'd like to know what is considered "commonly known".

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This question is too broad. Keep taking math classes and you will find out. Complex analysis uses neat techniques. –  glebovg Nov 20 '12 at 1:14
    
I think what you are referring to as 'bijective proofs' are what are commonly referred to as 'double counting arguments' in combinatorics. –  Tara B Nov 20 '12 at 1:25
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2 Answers

I can think of the following:

  • There are several interesting complex-analytic techniques for computing real integrals (think of residue calculus and methods of contour integration).
  • If you open some advanced books on statistical distributions and multivariate statistics (such as the Johnson, Kotz et. al. series on statistical distributions), you will find a plethora of clever integration techniques. This is due to the fact that integration is fundamental in probability and statistics (in fact, many questions in those areas reduce to integration problems).
  • Harmonic analysis and also methods from group theory that exploit symmetries of the underlying functions to get clever analytic integration results. (think of say integration over the orthogonal group.)
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This question is too broad. Keep taking math classes and you will find out. Complex analysis uses neat techniques. This and this might help. Also, Weierstrass substitution is one of my favorite techniques.


The class that uses almost all integration techniques imaginable is quantum mechanics, but I doubt you will see complex integration techniques in intro quantum.


If you have never seen Leibniz rule, you will be amazed the first time you see it.

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That is interesting. Where is integration used in quantum mechanics? (I am only a newcomer to the subject). –  Learner Nov 20 '12 at 1:30
    
When you need to compute expectation values and variances of various operators or taking Fourier transforms for example. Note that quantum mechanics is a physics class. –  glebovg Nov 20 '12 at 1:37
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