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:
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)?
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".