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We just learnt the different types of integration techniques in school such as substitution, by parts, etc. But, these methods seem kind of laborious. Do professional mathematics and theoretical physicists use these techniques itself or do they use more faster methods ? I would like to know these methods.

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Friendly-neighborhood CAS to the rescue. Take you pick. In a nutshell the integration methods don't get easier if you do more advanced stuff, just the range of integrals that are doable grows. Some integrals are only doable by advanced methods - see this for a taste. – Peter Sheldrick Dec 18 '12 at 12:27
so professionals use CAS only. They don't integrate manually ? Or do they use contour integration and other advanced stuff. Is it easier to integrate normal functions using contour integration ? – carboncopy Dec 18 '12 at 12:33
Are you teaching, trying to make some theoretical point, or just for plain fun? You try to integrate manually. Do you need to know some integral for research, job, life-&-death, etc. matters? You may try some of the numerous books with thousands of solved integrals there, or try some computer-based program (WA, Mathematica,Maple and etc.). – DonAntonio Dec 18 '12 at 12:47
Well, I want to know, since, I'm planning to take up something math related as a career. – carboncopy Dec 18 '12 at 12:49

I pretty much do everything with pen and paper if possible. This means I use all of the techniques I learned in school. For the types of problems I solve this typically results in problems that involve the stuff from 2nd semester calculus that you just learned. I use methods from complex variables occasonally and functional analysis stuff less occasionally. I also use software to check parts or the entire result when that becomes necessary (meaning checking).

I generally find that in order to determine if a technique or idea I choose is working or is likely to work, that I need to work things out. I learn things about the problem and my approach from the pen and paper method that I don't learn with the computer. On the other hand, sometimes when the computations are very tedious, repetative or just very complicated, I learn things by getting results with software. There is little if anything that I don't at some point do by hand.

Speed comes through experience not from having faster methods. Most of the integration techniques one learns in 2nd semester calculus become completely straightforward through repetition.

I suppose that is both the good news and the bad news. It becomes pretty easy after you do thousands (and probably thousands) of integrals. Hey, but you have to field a lot of ground balls to become a good shortstop.

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Oh ! That was kind of disappointing. I thought there would be one method to solve them all. But, thanks for the answer anyway. Guess, I need to practice some more ! – carboncopy Dec 20 '12 at 9:57

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