How many integration techniques are worth knowing?

[ozone]

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.

 

[tiny-tim]

hi ozone!

… 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.

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)​

 

[ZombieFeynman]

This trend of relying on CAS programs to do basic math worries me.

 

[AlephZero]

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.

 

[ozone]

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

$\text{dt} = \frac{\pm \text{dr}}{\sqrt{\frac{2}{\mu }\left(\text{Ecm} - U - \frac{l}{2\text{\mu r}^2}^2\right)}}$
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.

 

[tiny-tim]

hi ozone!

$\text{dt} = \frac{\pm \text{dr}}{\sqrt{\frac{2}{\mu }\left(\text{Ecm} - U - \frac{l}{2\text{\mu r}^2}^2\right)}}$

well, that's essentially dt = ∫ Ar(r² - B)^-1/2 dr

which is what i would call an obvious direct integral

 

[Jorriss]

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.

 

[alan2]

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.

 

[Angry Citizen]

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.

I fail to see the point of knowing advanced integration techniques when computers exist. This isn't the 1950s, and performing intensive calculations are sort of 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.

 

[ozone]

tiny-tim the integral is not that simple. There is an r-dependance in your potential such that your integral winds up looking like $\frac{2}{\tau }\int _{\text{Rmin}}^{\text{Rmax}}\frac{\text{dr}}{\sqrt{\frac{2}{\mu }\left[\text{Er}^2- \frac{\text{kr}^4}{2}-\frac{l^2}{2\mu }\right.}}\left(\frac{\text{kr}}{2}^3\right)dr$

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

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.

 

[tiny-tim]

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

 

[micromass]

I fail to see the point of knowing advanced integration techniques when computers exist. This isn't the 1950s, and performing intensive calculations are sort of 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.

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
$$\int \frac{x^3}{\sqrt{1-x^2}}dx$$
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.

 

[ZombieFeynman]

I fail to see the point of knowing advanced integration techniques when computers exist. This isn't the 1950s, and performing intensive calculations are sort of 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.

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.

 

[espen180]

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.

 

[cgk]

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.

 

[micromass]

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.

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?

 

[Vanadium 50]

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.

 

[cgk]

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?

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.

 

[Angry Citizen]

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?

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.

 

[micromass]

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.

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.

 

[Angry Citizen]

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.

I suspect it's faster to implement computational methods or table methods than to derive many of these integrals by hand. But I haven't needed to yet. None of the integrals I've been required to use so far in engineering undergrad have required more than integration by parts.

 

[micromass]

I suspect it's faster to implement computational methods or table methods than to derive many of these integrals by hand. But I haven't needed to yet. None of the integrals I've been required to use so far in engineering undergrad have required more than integration by parts.

Oh, you're an engineer. Yeah, then you'll probably be fine with using computers to solve your problems.

 

[romsofia]

I would say two techniques I would know on the back of my hand for physics: Know perturbation theory and how to find series solutions to hard integrals. These two techniques might not help with EVERY problem, but knowing them helps a lot, my opinion anyway.

Good luck.

 

[WannabeNewton]

Hi there ozone! In my current classical mechanics class we too get a large number of messy integrals to compute in problem sets. In my experience they have always reduced to integrals where standard calc 2 techniques (sub., parts, trig sub., partial fractions, etc.) have sufficed in finding the solution. The hard part might be in finding out how to reduce it to a familiar form so as to get an expression involving elementary functions but it is all practice in the end. I would say don't listen at all to what Angry Citizen or cgk are saying; integration is an art and it is very enjoyable in evaluating the pedagogical integrals presented in problem sets.

 

[ballistikk]

At the very least, know the techniques from Cal II, plus Stokes, Green's, and Residue Theorem.

Just recently, I had to drag out the Mean Value Theorem to solve a delta function problem.

 

[FalseVaccum89]

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?

Exactly. What I don't understand is, how can people encounter a new math concept and not want to know how it works? :P

 

[Best Pokemon]

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.

Some guys in this forums found integrals that Mathematica couldn't compute: http://www.wilmott.com/messageview.cfm?catid=10&threadid=85389

 

[D H]

Oh, you're an engineer. Yeah, then you'll probably be fine with using computers to solve your problems.

Hey! I resemble that remark!


I completely agree with the concern about over-reliance on computer algebra systems to solve simple problems. And to some extent, complex ones.

When Mathematica or Maple or Maxima spews out some incredibly long answer, how are you going to use it? You can't use it in a computer program; that ridiculously long equation is just too expensive computationally and almost inevitably suffers huge precision loss problems. You can't use it by hand; it's just too long. It's worthless. So you solve the problem by hand, and come up with a nice, compact expression. That ridiculously long answer from Mathematica (or whatever) is just that -- ridiculously long.

Some guys in this forums found integrals that Mathematica couldn't compute: http://www.wilmott.com/messageview.cfm?catid=10&threadid=85389

Normally we don't allow references to other forums, but this response in that thread is just awesome:

In my experience Mathematica is awesome, except with the specific problems I try. I always end up doing it with pen and paper after 30 tries.​

 

[Angry Citizen]

When Mathematica or Maple or Maxima spews out some incredibly long answer, how are you going to use it? You can't use it in a computer program; that ridiculously long equation is just too expensive computationally and almost inevitably suffers huge precision loss problems.

Just numerically integrate the integral itself.

I would like to see some examples where Mathematica spits out long, complicated solutions to integrals that cannot then be converted to a simpler form, when the same integral can be made into a nice and neat solution by hand. I've never used Mathematica itself, but I know MATLAB has a command "simple" that spits out about a dozen and a half alternative forms to a given algebraic expression. Usually one of them is the one you want.

 

[ozone]

Sorry I've been caught up in school and did not have time to post. I guess that I will put some time into integration techniques when I can, and thank you for the constructive comments everybody.

 

[MathWarrior]

I would say you need to know all the integration techniques to some degree, basic ones you encounter in calculus 1-3 definitely. This however depends entirely on what your major is, and what you intend to do.

 

[cgk]

Some guys in this forums found integrals that Mathematica couldn't compute: http://www.wilmott.com/messageview.cfm?catid=10&threadid=85389

That's not hard to come up with. Just write down a horrible contrived expression, take its derivative, and--tada--you got an integral you know the answer to but no one else (human or machine) has any chance of ever calculating.

The question is not whether there are expressions for which a--possibly simple--closed-form analytical expression of its integral exists. Of course there are. The question is: If you do not happen to be able to already know or be able to guess the answer, do you, as a human, have any chance of coming up with an algorithm to get the answer which Mathematica cannot? And this can usually happen only if you have some additional information about the problem (e.g., symmetries, magic coordinate transformation, etc) which Mathematica does not have. Because it sure *DOES* know how to apply all the basic integration techniques, and it does know them much better than you as a human do.

Although I have do admit, I was quite surprised by the BesselJ example. I'll try this some time (maybe some assumptions were missing, and the posted identity doesn't always hold).

btw: I'd bet several of the examples on that page would just work in Mathematica if you turned on the cubic and quartic radicals (which are disabled by default because calculating with them might stress your patience). And the fact that you can just do something like this: Ask the program to solve with techniques you could never hope to apply by hand, is already quite awesome. (actually, most of the symbolic integration techniques one could never apply by hand. Some of them are really cool.)