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How many integration techniques are worth knowing?

  1. Oct 26, 2012 #1
    I will make this post quick, and if anyone can weigh in with their thoughts it will be appreciated.

    Right now I am taking classical mechanics at my university, and I have noticed a trend of very difficult integrals on our problem sets. Our most recent one required 4-5 substitutions to yield the correct answer. By the way some of these substitutions were completely unintuitive, and I feel that I would have never been able to have derived them.

    I never bothered worrying much about learning integration tricks since I figured wolframalpha/mathematica were sufficient at solving all the integrals I would ever see..but its becoming obvious that this just might not be the case.

    I'm wondering what the best way to go about learning some integration tricks would be, and if it is really worth it in the long run
    Note * I don't think my teacher will be including crazy math problems on our exams so I will only bother learning these integrals if it is worth it for my scientific pursuits*

    Thank you.
     
  2. jcsd
  3. Oct 27, 2012 #2

    tiny-tim

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    hi ozone! :smile:
    i'm surprised it should take that long … i suspect there was a short-cut

    can you show us (or link to) the solution?

    (or maybe what you call a substitution, i would call an obvious direct integral? …

    eg ∫ 4cosθsin3θ dθ is obviously sin4θ + C)​
     
  4. Oct 27, 2012 #3

    ZombieFeynman

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    This trend of relying on CAS programs to do basic math worries me.
     
  5. Oct 27, 2012 #4

    AlephZero

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    If it's taking you "4 or 5 substitutions" to do homework questions, there's a short answer to "how many techniques are woth knowing":

    More than you know already.
     
  6. Oct 27, 2012 #5
    I agree I could stand to learn more about integration.. could you point me in the right direction?

    Anyways the problem was:

    Show that the areal velocity is constant for a particle moving under the influence of an attractive force given by F = -kr. Calculate the time averages of the kinetic and potential energies and compare with the the results of the virial theorem.

    In order to check this we had to use the differential orbit equation and eventually we came to

    [itex]\text{dt} = \frac{\pm \text{dr}}{\sqrt{\frac{2}{\mu }\left(\text{Ecm} - U - \frac{l}{2\text{$\mu $r}^2}^2\right)}} [/itex]
    we could take this equation and back sub it into our time integral of our potential, from which we could change the bounds so that they were in terms of our radius min or max.. However the integration was pretty nasty!

    If your not familiar with the variables in this equation I'm sorry.. our teacher hasn't posted a solution key online yet.
     
  7. Oct 27, 2012 #6

    tiny-tim

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    hi ozone! :smile:
    well, that's essentially dt = ∫ Ar(r2 - B)-1/2 dr

    which is what i would call an obvious direct integral :wink:
     
  8. Oct 27, 2012 #7
    For undergraduate coursework, you need all the integration techniques one learns in calc II, plus a bit more. For graduate coursework, those old techniques are mainly just used to simplify integrals and then ultimately either integral transforms, residues or green's functions are used to give a problem that final blow. For actual research, I wouldn't know.
     
  9. Oct 27, 2012 #8
    The more you understand the better you will be at whatever you end up doing. I think that the use of symbolic math programs should be abolished along with graphing calculators to teach elementary mathematics.
     
  10. Oct 27, 2012 #9
    I fail to see the point of knowing advanced integration techniques when computers exist. This isn't the 1950s, and performing intensive calculations are sorta their point. I mean, do you do numerical methods by hand? Yeah, probably not.

    OP: Unless you can reasonably expect to see these kinds of integrals on closed book, closed note exams, I wouldn't bother with them. There is literally no reason to know them.
     
  11. Oct 27, 2012 #10
    tiny-tim the integral is not that simple. There is an r-dependance in your potential such that your integral winds up looking like [itex]\frac{2}{\tau }\int _{\text{Rmin}}^{\text{Rmax}}\frac{\text{dr}}{\sqrt{\frac{2}{\mu }\left[\text{Er}^2- \frac{\text{kr}^4}{2}-\frac{l^2}{2\mu }\right.}}\left(\frac{\text{kr}}{2}^3\right)dr[/itex]

    since [itex]\text{Since } <U>\text{ }= \frac{1}{\tau }\int Udt[/itex]
    and [itex]\text{where } U = \frac{\text{kr}}{2}^2[/itex]

    My point is that there was very little chance I could ever figure out this integral, and I just want to know if I should be worried.
     
  12. Oct 27, 2012 #11

    tiny-tim

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    ok, so it's now A ∫ r3(B2 - (r2 - C)2)-1/2 dr

    = A ∫ r(r2 - C)(B2 - (r2 - C)2)-1/2 dr + A ∫ Cr(B2 - (r2 - C)2)-1/2 dr

    which is the sum of two obvious direct integrals :wink:
     
  13. Oct 27, 2012 #12

    micromass

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    This response is pretty sad. And it is indicative of a modern evolution to rely more computers. This is an evolution that I absolutely do not like.

    Of course, when I'm given an annoying integral or calculation, I will resort to a computer. The difference is that I know how to do the integral or the calculation. The only reason that I don't do it, is that it would take a long time and that I would make errors. This is the only advantage a computer should have: that it is more accurate and it is faster. The advantage of a computer should not be that he knows more techniques.

    I think it is a pretty sad evolution that people don't know how to do standard integrals like
    [tex]\int \frac{x^3}{\sqrt{1-x^2}}dx[/tex]
    People should know how to do those integrals even if it were just out of self-respect.

    One very major gap in my education are differential equations. I was never really required to solve those things, so I don't know how to. And now whenever I see a differential equation, I feel very uncomfortable because I don't know how to handle them. Sure, a computer can solve them for me, but I don't feel alright with that. I never ever need differential equations right now in my research, but I still feel bad about not knowing them.
     
  14. Oct 27, 2012 #13

    ZombieFeynman

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    This outlook is foolish, in my opinion. Computer algebra systems are useful when one already knows all the techniques and tricks. Ones time as a student is the time to learn these things so when they do research later they can double check their computers calculations when then need to.
     
  15. Oct 27, 2012 #14
    Last time I checked, every time an important calculation in engineering is made on a computer, for example the maximum load a certain support admits before breaking, is double-checked by hand. So yes, you need to know the integrals.
     
  16. Oct 27, 2012 #15

    cgk

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    I wholeheartedly second Angry Citizen's opinion. I see little reason in becoming good at doing things a computer can do *much better* than you ever could anyway. It is much more worthwhile to gear your efforts into (1) learning how to make the computer do them, (2) teaching it how to do things it currently cannot do. After all, all knowledge comes with a price: If you invest time in learning integration techniques, you lose that time for learning something more useful.

    And integrals... well. The basic rule is: If Mathematica cannot do an integral (after reasonable subsitutions on your side), then unless you by chance know some very obscure trick or you are an expert in hypergeometric functions, then your chance of getting it right with pen and paper are close to zero.

    And I actually work in a numerical branch of physics. I regularly deal with analytic integrals. And since I was done with coursework, I was *never once* inclined to do any integral with pen and paper alone.
     
  17. Oct 27, 2012 #16

    micromass

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    And how are you going to do that if you don't know the techniques yourself? How are you possibly going to learn how to make a computer do an integration if you can't do it yourself?
     
  18. Oct 27, 2012 #17

    Vanadium 50

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    It's also often necessary to understand what the integral evaluates to to follow a derivation.

    Getting back to the OP's question, I would say between 150 and 200, possibly more. Which is why usually this is not taught as a list of integrals to memorize.
     
  19. Oct 27, 2012 #18

    cgk

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    First, knowing how to program a technique and being good at applying a technique are two completely separate things. For example, I know how a floating point multiplication circuit works, and I could easily program a 128bit floating point type myself, but I actually use Python to multiply two-digit numbers.

    Second, I was unclear on that: When I said "make the computer do things it currently cannot" I was not referring to integration techniques. As I said, I think the integration problem is solved to a degree that you can't do anything on it unless you are an legitimate expert on this particular field (and even then maybe not). I mean a more general approach to learning computing: Getting better at symbolic algebra, programming, etc. There are plenty of things computers can't do easily, and to make them do that, you need to know programming and advanced techniques in specialized fields (e.g., chemical physics, data analysis, finance...). These skills are much more valuable than getting good at doing basic techniques you will never need in practice.
     
  20. Oct 27, 2012 #19
    I can't for the life of me remember a thing about trig substitution or how to deal with radicals in the denominator when integrating. However, the fact that I have a pulse and an IQ somewhat higher than 100 means I'm capable of finding a Stewart book, flipping to the required page, and spending ten minutes learning how to do the technique. There is no need to know how to do it. It is sufficient even for inefficient hand derivations to know that these techniques exist. Alternatively, table of integrals.
     
  21. Oct 27, 2012 #20

    micromass

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    Great!! I wish you the best of luck in grad school. If I have to keep looking up things like that, I wouldn't get to research at all.
     
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