# How can I devise a general approach to solving an indefinite integration problem?

INTRODUCTION:

I'm trying to create a general approach in solving integration problems around 7 specific methods.

Basic Formulas, Substitution, Numerical Integration, Integration by Parts, Integration using trigonometric rules, Integration using trigonometric substitution and Integration Tables.

In my CALC_2 class we blew through all this in about 2 weeks so I'm having a hard time organizing my thoughts. I've been struggling to find out how to construct a flowchart directing me to a specific method based on the parameters of the problem.

Therefore, in creating an organized approach with key points directing me to specific methods, it would serve as a good tool for me on reiterating integration entirely.

QUESTION:

I need a push in the right direction that would help me at least get started. I have these seven methods laid out in front of me and I can't seem to find a good starting point. Any advice and/or resources you might have in helping me configure my analysis would be greatly appreciated. I'm not asking for a complete layout, as that would defeat the purpose of this project. Just some good advice would help me out greatly.

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I suggest you think about what each of those problems looks like, i.e. if you're integrating a product where one term looks like the derivative of the other it should push you to try a substitution, if you are integrating a product of two unrelated functions perhaps it should be tackled with parts, if you have a term that looks like $\sqrt{a^2 - b^2}$ you should recognise Pythagoras and try a trig substitution. – in_wolframAlpha_we_trust Jun 4 '13 at 8:35
Most textbooks have a section on general integration strategy. The sections tend to be long and with rather general advice. The best strategy is just to do many problems from each section before you mix all methods. Make sure you learn the very basic examples very well. And just hang in there! – Maesumi Jun 4 '13 at 8:46
I agree with Maesumi, really there isn't any substitute for general practice in integration. To this day when I tutor students when they have trouble with integration and I point them in the right direction its really (mostly) due to many years or doing calculus. I've done so many integrals it starts to become a habit to recognize certain signs like a function and its derivative, products, rational functions, etc. – Dan Jun 4 '13 at 9:08

I quite understand your frustration (from one electronics engineer to another), but you'd probably have to look at Mathematica's internal code to see how they do it. The way I teach my high school students is to start with the table of standard integrals (pattern recognition), then to look for a possible substitution which might simplify matters, then to aim for an f-dash over f expression. This isn't the end of the story, but might get you started.

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I think Mathematica (and most symbolic packages) use the Risch algorithm. en.wikipedia.org/wiki/Risch_algorithm – in_wolframAlpha_we_trust Jun 4 '13 at 8:46
However, presumably the algorithms Mathematica employs for integration would be very different from those a human could (feasibly) carry out (the Risch algorithm linked above is very different from that taught in a standard class!) I'm not so sure looking at the code would help here, unless Shane is hoping to program some implementation of this. – WeierstrassSauce Jun 4 '13 at 8:47
Yes, the Risch algorithm is not human-friendly. – in_wolframAlpha_we_trust Jun 4 '13 at 8:54
@in_wolfram The non-algebraic-function case of the Risch algorithm is just simple rational function computations. Given a random integral, one will have much better luck using the algorithm than ad-hoc techniques. But given an integral constructed by a human to be solved by ad-hoc techniques (e.g. exercises, contest problems, etc), the algorithm may be more tedious. – Key Ideas Jun 4 '13 at 22:10