9

I've been pretty frustrated lately with my poor integration skills. A lot of times I find that in my math or physics classes I understand the concepts behind a question, reduce it to an integral, and then find myself unable to solve it. Yet some people both on this board and at my university are deft hands at solving complicated integrals with a variety of tricks. I'd really like to get better at integrals and start approaching their level. I was hoping that there might be suggestions of workbooks or textbooks that are designed purely on increasing the reader's ability to solve difficult integrals, and also suggestions on a good general philosophy to take to get better at this as well.

Also, I'm unsure if this is the wrong place to ask this question. If so, I apologize and I'll remove the question.

Mark Viola
  • 179,405
Bookie
  • 367
  • 3
  • 12
  • 1
    I like this question and I think alot of students/mathematicans have asked themself the same question. My professor told me: to differentiate is easy since you "only" have to use different rules like product, sum etc rule - but integration is an art since you never know how to because even nicely looking integrals can turn out to be super ugly. –  Feb 10 '17 at 15:49
  • 1
    Instead of looking for a book I think it could be very interesting to collect different methods. Maybe this has already been done :) –  Feb 10 '17 at 15:51
  • 1
    for some classes of integrals there exist canonical procedures to solve them. for example integrals of rational functions of trigonometric expressions can be reduced to integrals of rational functions via Weierstrass subsitutionen. These in turn can be solved canonically by partial fraction decomposition... Learning this cases where a "simple" algorithm exists save you a lot of work – tired Feb 10 '17 at 16:17

2 Answers2

2

I'd have to agree with the other answer, I do not consider myself to be an integration master, nowhere near one, however your "basic" (avoiding things like residue and other crazy stuff) difficult integrals I can do fairly easily. I saw a certain beauty to an integral which motivated me from going to your basic reverse power rule to u subs, some interesting trig substitutions, and I would practice with those. Once you can handle those you can escalate, I liked looking for difficult integrals and then looking at how someone else solved them to add a new technique to my toolbox. So to sum all this up, start small and then master the little things, from there try and like google "really hard integrals" and see if you can see how those are done and then practice with more difficult stuff. (When you're ready for the big leagues MIT has an amazing integration bee whose problems and solutions are online)

Teh Rod
  • 3,108
  • Second the recommendation for the MIT Integration Bee! – John Hughes Feb 10 '17 at 17:24
  • @JohnHughes when I first started that's where I went after I had the basics and I could like not even do one in under 10 minutes. Now I can do like almost all of the ones on the more recent integration bees – Teh Rod Feb 10 '17 at 17:25
  • "residue and other crazy stuff" hum hum. The method of residues is among the most efficient, and furthermore has beautiful aspects. Of course, this is beyond what the OP asks: he/she needs advise in real analysis methods – Jean Marie Feb 10 '17 at 17:45
  • @JeanMarie oh really didn't know that, I'm actually trying to further build a base in complex analysis to work up to the residue theorem – Teh Rod Feb 10 '17 at 17:46
1

I suggest doing the exercises in Spivak's Calculus, in the "techniques of integration" chapter. When you've done all of those, you'll be a pretty solid integrator. (BTW, almost none are easy). I guess the short form of this is "practice makes perfect, but only if you practice stuff you don't already know."

John Hughes
  • 93,729