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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".

  • 0
    Look at http://en.wikipedia.org/wiki/Differentiation_under_the_integral_sign, http://math.stackexchange.com/questions/12909/will-moving-differentiation-from-inside-to-outside-an-integral-change-the-resu, http://math.stackexchange.com/questions/9402/calculating-the-integral-int-0-infty-frac-cos-x1x2-mathrmdx-wit#9409, and http://www.math.uconn.edu/~kconrad/blurbs/analysis/diffunderint.pdf.2012-11-20
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    This question is too broad. Keep taking math classes and you will find out. Complex analysis uses neat techniques.2012-11-20
  • 0
    I think what you are referring to as 'bijective proofs' are what are commonly referred to as 'double counting arguments' in combinatorics.2012-11-20

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