# Why learn integration techniques?

• ice109
In summary, most integrals aren't integrable to and can be evaluated numerically to any degree of accuracy, so why bother learning these esoteric techniques? However, when it comes to proving theorems, computers are rather useless in many cases, so why bother learning calculus?

#### ice109

if most integrals aren't integrable to and can be evaluated numerically to any degree of accuracy why learn these esoteric techniques at all?

Why bother learning any maths if a computer can solve it all numerically...?

matt grime said:
Why bother learning any maths if a computer can solve it all numerically...?
Math isn't just about finding solutions to equations - when it comes to proving theorems computers are rather useless in many cases. And even when computers can help us prove theorems - it's only when there're to many special cases for a person to examine, the four color theorem for example.

Why learn how to differentiate when taking a very close finite difference will do?

ice109 said:
if most integrals aren't integrable to and can be evaluated numerically to any degree of accuracy why learn these esoteric techniques at all?
Be careful here. For example, it is often claimed that $$e^{-x^2}$$ is not integrable. That of course is not true. The integral of this function is very well-known:
$$\int_0^x e^{-t^2}dt = \frac{\surd \pi} 2 \text{erf}(x)$$

When mathematicians say something isn't integrable what they really mean that the solution cannot be expressed in terms of some limited set of functions, typically the elementary functions. If an integral comes up often enough mathematicians (or physicists, or whoever) will define a function based on this integral. The error function is one such special function.

That one has to resort to numerical techniques to solve a numerical problem is not limited to the special functions. What are the exact values of $\surd 2$ and $\sin 1$? We have to use numerical techniques to evaluate $\surd 2$ and $\sin 1$, even though both the square root and sine functions are elementary functions.

No, integrability means excatly what it means, that functions' riemann sum converges, and the limit of this is called riemann integral.
a function which isn't integrable in a specific domain, means that its riemann sum doesn't converge.

at least this is one way of defining integrability.

Loop, while that is one way of defining integrability, it obviously is not the sense meant by the original poster. What the OP really meant was (my changes to the OP are in italics):

"if most antiderivatives cannot be expressed in terms of elementary functions but can be evaluated numerically to any degree of accuracy why learn these esoteric symbolic integration techniques at all?"

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D H said:
Loop, while that is one way of defining integrability, it obviously is not the sense meant by the original poster. What the OP really meant was (my changes to the OP are in italics):

"if most antiderivatives cannot be expressed in terms of elementary functions but can be evaluated numerically to any degree of accuracy why learn these esoteric symbolic integration techniques at all?"

yes that was a mistake on my part.
D H said:
That one has to resort to numerical techniques to solve a numerical problem is not limited to the special functions. What are the exact values of $\surd 2$ and $\sin 1$? We have to use numerical techniques to evaluate $\surd 2$ and $\sin 1$, even though both the square root and sine functions are elementary functions.

yes i don't see anything odd about using a numerical method for a numerical problem such as evaluating a definite integral.

this begs the question: what are antiderivatives used for other than elegantly evaluating definite integrals.

actually nm that question cause I'm sure there exists a use somewhere, maybe proving certain things or some such thing.

but calculus classes are for engineers, why do they need to learn these techniques? I'm not an engineering student i don't know but i would guess their integrals are simple? even if they are why waste their time when everyone these days has access to some way of evaluating them numerically.

Gib Z said:
Why learn how to differentiate when taking a very close finite difference will do?

that's not a very good rebuke because i don't need to resort to finite differences unless I'm taking discrete data which most people don't do.

before i get a lot of people rebuking me note that i think the ideas behind the indefinite/definite integral and derivative are very important, it's the rigamarole i don't see the necessity of.

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ice109 said:
that's not a very good rebuke because i don't need to resort to finite differences unless I'm taking discrete data which most people don't do.

I don't get what you mean by having "to resort to" them, my point was that taking close finite differences can give us the numerical value for the derivative at that point to any degree of accuracy, just as numerical integration techniques do for integrals. I thought this parallel might have made the answer a tiny bit easier to see, but it obviously doesnt, my bad :(

My point was, exact answers are always nice =] And when we can't get an exact answer, call it something new.

D H said:
Loop, while that is one way of defining integrability, it obviously is not the sense meant by the original poster. What the OP really meant was (my changes to the OP are in italics):

"if most antiderivatives cannot be expressed in terms of elementary functions but can be evaluated numerically to any degree of accuracy why learn these esoteric symbolic integration techniques at all?"

And I think that's a good question, to wit I have a bad answer (*cough* speculation *cough*).

I think that many students haven't fully mastered algebra when they begin learning calculus. They can do algebra, but they can't think with algebra. In the process of doing all those limits, derivatives and integrals they obviously gain an intuition for calculus, but they also only then truly become adept at thinking with algebra. And that is crucial for doing well in physical science and engineering.

I say this because my students at the beginning of the year had no problem doing algebra, but they really struggled with interpreting and understanding algebraic equations even when they understood the physical concepts. By the end of the year, that really wasn't a problem. Now certainly you can attribute it to both physics and calculus, but I have a feeling that calculus played a stronger role in that learning process.

Integration is not always about finding the numerical value of an integral. Just pick up any book on pure mathematics. You'll most likely find integrals on every page, but not one of them will be evaluated to give a numerical answer. In fact, you can even just look at any book on physics or engineering. Most results will involve symbolic integration as intermediate steps in the derivations.

Gib Z said:
I don't get what you mean by having "to resort to" them, my point was that taking close finite differences can give us the numerical value for the derivative at that point to any degree of accuracy, just as numerical integration techniques do for integrals. I thought this parallel might have made the answer a tiny bit easier to see, but it obviously doesnt, my bad :(

My point was, exact answers are always nice =] And when we can't get an exact answer, call it something new.

and what is exact to be exact? is $\sqrt{2}$ exact? sure but who cares because you can't do anything with that symbol except algebraic manipulation. i would say derivatives are useful because we can always take a derivative of a continuous function and it's quicker than finite differences, that was my point.
dx said:
Integration is not always about finding the numerical value of an integral. Just pick up any book on pure mathematics. You'll most likely find integrals on every page, but not one of them will be evaluated to give a numerical answer. In fact, you can even just look at any book on physics or engineering. Most results will involve symbolic integration as intermediate steps in the derivations.
i'm going to get lynched for this one but that again is a completely pointless thing I am my humble opinon. anyway then symbolic integration is a tool for research engineers and physicists not practicing engineers. meaning of course i had in mind all the phenomenology that goes on in physics when i asked this question hence i restricted it to engineering. i guesss i should've been more specific about what kind.

ice109 said:
i'm going to get lynched for this one but that again is a completely pointless thing I am my humble opinon. anyway then symbolic integration is a tool for research engineers and physicists not practicing engineers. meaning of course i had in mind all the phenomenology that goes on in physics when i asked this question hence i restricted it to engineering. i guesss i should've been more specific about what kind.

If you don't understand how integration works, why should you have any reason to expect that the number your computer is giving you will be accurate? Black boxes that give you magical answers can be dangerous things. If you don't understand how the program does things, you run the risk of getting an erroneous answer out of it. This might not be a large problem if you're a research scientist just trying to solve an integral to use in a calculation you're doing for a paper, but if you're an engineer and a computer gives you the wrong answer you could end up with a collapsing bridge. In order to understand how programs do integration, you need to know how integration works - convergence of the integral, how many bins you need to accurately represent the area under the curve, etc.

Furthermore, there could be issues of efficiency. Some integrals can be transformed into other integrals which could be easier to solve numerically. If you're writing your own integration problem you need to know how to work with integrals if you're going to turn the integral you have into something nicer to evaluate numerically.

Lastly, what if the integral you need to do is a simple one, or one with problem points that might cause a computer grief due to singular points that are easily dealt with symbolically? Why waste time getting a program to numerically solve $\int_a^b dx~\ln x$ when you could easily find the integral to be $\left[x \ln x - x\right]_a^b$? The computer would have a much easier time evaluating [b ln b - b] - [a ln a - a] than summing up several bins.

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ice109 said:
i'm going to get lynched for this one but that again is a completely pointless thing I am my humble opinon. anyway then symbolic integration is a tool for research engineers and physicists not practicing engineers. meaning of course i had in mind all the phenomenology that goes on in physics when i asked this question hence i restricted it to engineering. i guesss i should've been more specific about what kind.

You clearly have no idea what engineering is. Post this in the engineering forum and see what they say.

dx said:
You clearly have no idea what engineering is. Post this in the engineering forum and see what they say.

how about instead of me double posting you give me an example since that is the point of this thread.

Look I'm not a polemicist, let everyone keep that in mind. For what it's worth I'm a pure math student so I'm not some lazy dunce trying to argue their way out of learning abstract concepts.

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ice109 said:
For what it's worth I'm a pure math student so I'm not some lazy dunce trying to argue their way out of learning abstract concepts.

The truth is that we teach integration techniques so that we can employ professors into mathematical research en masse.

If you want a better answer (in terms of morality, but only equally true in the world), calculation is part of the mathematical tradition and without it pure mathematics would collapse. The problem is that now that you don't care about calculations, you are one step closer to not caring about theorems. As you can see by looking at history, the criteria for what is a theorem and what is a mere example always shift in a more jaded direction over time --- towards the view that more and more is trivial --- but if we follow this trend to its logical conclusion we see that math will die of the same snob-strangulation that kills technique in other fields.

Say you have to evaluate this integral 100,000 times for various values of "a" (this is a completely realistic scenario if you are simulating something)

$$\int_0^a \frac{1}{x^{0.999}}dx$$

which do you think is faster: Waiting for some numerical algorithm to converge, or to do it by hand?

Or what about a function that oscillates like crazy? It takes a while for that one to converge too.

I think a much more interesting question would be "why bother learning integration techniques when Mathematica can usually symbolically evaluate integrals for you?"

In my opinion, teaching the techniques allowed me to appreciate how technology shortened the many many integrals that I have had to do by hand.(And most of those, were long and nasty looking!)

daudaudaudau said:
Say you have to evaluate this integral 100,000 times for various values of "a" (this is a completely realistic scenario if you are simulating something)

$$\int_0^a \frac{1}{x^{0.999}}dx$$

which do you think is faster: Waiting for some numerical algorithm to converge, or to do it by hand?

Or what about a function that oscillates like crazy? It takes a while for that one to converge too.

Actually in your case doing it by hand is still slower because you still have to evaluate the antiderivative 100,000 times. Simply use an algorithm that's quadrature based on rational functions (as opposed to polynomials) in C code and a home pc will beat you to the punchline. Even if you allow yourself the use of a scientific calculator when you evaluate that function 100,000 times!

DavidWhitbeck said:
Actually in your case doing it by hand is still slower because you still have to evaluate the antiderivative 100,000 times. Simply use an algorithm that's quadrature based on rational functions (as opposed to polynomials) in C code and a home pc will beat you to the punchline. Even if you allow yourself the use of a scientific calculator when you evaluate that function 100,000 times!

Obviously I was going to evaluate the antiderivative on a computer. I'm not doing simulations with pen and paper.

daudaudaudau said:
Obviously I was going to evaluate the antiderivative on a computer. I'm not doing simulations with pen and paper.

BUT if you were kidnapped and forced to do it by hand, you would know how to!

Pure mathematics must aim to get exact answers in the most elementary form possible, without the aid of anything other than pen, paper, and ones mind.

Meh, its a shortcut for proving lots of properties of the integral. I mean, you could prove that integral(2*x) is even, nonnegative, has only 1 zero at 0, can be described by a focus and directrix, has no inflection points, describes the path of a projectile, etc etc all from the definition of the integral. On the other hand, you could note that it equals x^2 and then all the properties become trivially obvious.

Gib Z said:
Pure mathematics must aim to get exact answers in the most elementary form possible, without the aid of anything other than pen, paper, and ones mind.

you're young and naive so it's ok but math is not this exalted thing you believe it is. it's just another game whose rules change and evolve with the times.

daudaudaudau said:
Say you have to evaluate this integral 100,000 times for various values of "a" (this is a completely realistic scenario if you are simulating something)

$$\int_0^a \frac{1}{x^{0.999}}dx$$

which do you think is faster: Waiting for some numerical algorithm to converge, or to do it by hand?

Or what about a function that oscillates like crazy? It takes a while for that one to converge too.

You don't seem to understand that Mathematica can evaluate the symbolic antiderivative, and so your argument is utterly moot.

You can try a slice of this technology at http://integrals.wolfram.com/index.jsp"

Furthermore, Mathematica can evaluate all the anti-derivatives that humans have ever done as well as many more. It can also solve all the ordinary differential equations that can be solved by hand.

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Gib Z said:
Pure mathematics must aim to get exact answers in the most elementary form possible, without the aid of anything other than pen, paper, and ones mind.

Then a more descriptive term than "pure" would be "anthropocentric."

I for one think that your sense of the word "pure" is a narrow-minded and destructive attitude that has setback mathematics tremendously since it was first espoused by G.H Hardy.

One only has to look at the previous generation of great mathematicians who Hardy did not corrupt, for example in the writings of Bertrand Russell we see that he considers the joy of mathematics to be the fleeting transcendence of the human condition.

Has a computer ever solved an integral in closed form that a human cannot?

Sure I understand that numerically, computers can do stuff we can't. This question is more aimed at asking if computers have discovered anything new. As in we posed it a mathematical question which we did not know the answer too, it gave us an answer which was later verified to be correct.

Eidos said:
Sure I understand that numerically, computers can do stuff we can't. This question is more aimed at asking if computers have discovered anything new. As in we posed it a mathematical question which we did not know the answer too, it gave us an answer which was later verified to be correct.
Doesn't the four colour theorem fall into such a category?

Gib Z said:
Pure mathematics must aim to get exact answers in the most elementary form possible, without the aid of anything other than pen, paper, and ones mind.

Pure mathematicians can prove any theorem they want with only a pen and paper! Pure mathematicians find counterexamples ALL the time and don't even think twice about it. I heard there was this pure mathematician who was eating at a diner. And when some dude started using a calculator, the mathematician killed the whole town. My friend Mark said he saw a pure mathematician totally prove the riemann hypothesis just because a kid forgot the constant of integration.

Facts:
1) Pure mathematicians are mammals.
2) Pure mathematicians can recite pi - backwards.
3) Pure mathematicians have never climbed Mt. Everest. It is too easy. When pure mathematicians want to go mountaineering, they construct the dirac delta distribution out of a pencil, 4 sheets of paper, and a large amound of duct tape.

And that's what I call REAL ULTIMATE MATHEMATICS.

If you don't believe that pure mathematicians have REAL ULTIMATE POWER you better get a textbook right now, or they will blow your mind up! It's an easy choice, if you ask me.

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Perhaps you all interpreted too much emphasis of the pen/paper bit. I don't personally like proving theorems with computers, like the 4 color theorem, but that's another argument. I was wanted to state that basically its *nice* to have things evaluated in exact terms, and in the most elementary form possible. Everyone knows how to simplify algebraic expressions and the like, we can state the same thing in many ways, but we want the simplest one to work with. Same one applies here. Thats all I wanted to say.

@defennder: I suppose you are right. The four colour theorem does fall into that category.
What I meant to say was that computers cannot solve original problems all by themselves. Someone had to program in the routine used.

@maze: That was awesome! :D
Thanks for the laugh.

Hard Cold Logic

Here is an exerp from Wikipedia. "Euclidean geometry is a mathematical system attributed to the Greek mathematician Euclid of Alexandria. Euclid's text Elements is the earliest known systematic discussion of geometry. It has been one of the most influential books in history, as much for its method as for its mathematical content. The method consists of assuming a small set of intuitively appealing axioms, and then proving many other propositions (theorems) ."

The same question of "utility" has been directed to the teaching of Euclidean geometry in high school. I think the utiliterians prevailed on this one and computer assisted learing products such as The Geometers Sketchpad have prevailed to the exclusion of the hard cold logic necessary to follow a set of axioms to a correct conclusion.

I'm not an educator so maybe someone in the education community can amplify these comments.

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jimvoit said:
The same question of "utility" has been directed to the teaching of Euclidean geometry in high school. I think the utiliterians prevailed on this one and computer assisted learing products such as The Geometers Sketchpad have prevailed to the exclusion of the hard cold logic necessary to follow a set of axioms to a correct conclusion.

I'm not an educator so maybe someone in the education community can amplify these comments.

That's terrible! High school geometry was what turned me onto math and science. I loved doing those proofs, you know where you had to do those constructions on one side, and list the propositions, axioms etc on the other side line by line to arrive at the logical conclusion that you wanted to show. I didn't love math before that class, but after I knew that I wanted to go into either physics or math. If it wasn't for that class, I might have ended up in the humanities instead!

Anyway that explains alot. My students had no concept of logical argumentation, and even in calculus were taught to "prove" an identity by assuming it and working it through to 1=1. Sigh.

ice109 said:
yes that was a mistake on my part.
but calculus classes are for engineers, why do they need to learn these techniques? I'm not an engineering student i don't know but i would guess their integrals are simple? even if they are why waste their time when everyone these days has access to some way of evaluating them numerically.
te integral and derivative are very important, it's the rigamarole i don't see the necessity of.

Analytic solutions offer insight. Numerical answers do not.

I would say, yeah... as a student, we deal with simple integrals. I mean, really, what else would we get? Would it be realistic to give a integral that requires some type of numerical method to solve it for? I have obviously had homework, questions, but not on a test.

As an example, one may use linear approximations to complicated expressions and then solve these simplified expressions. Often times these simplified expressions are valid enough, and they offer loads more insight into a problem than a sequence of numbers.

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Many times higher order terms are chopped out so things can be formed into nice algebraic solutions for a first order approximation. This gives someone a feel of the problem, and then when the accuracy is required we move onto using a nice numerical routine to include all the "hard" terms. As an example, one can reduce systems that have cross coupling terms as they could be negligible for a first order approximation. Another example is dealing with transistor models, the "real" models (which sim software such as Cadence uses) is way more complicated then the simple expressions engineers come to memorize. However, these simple expressions allow us to get an idea if the numbers being spit out of the computer are within a reasonable measure of accuracy.

Also, if we didn't teach basic methods how would you even test a numerical method you write on the computer? The first thing I do when writing a method from scratch is plug in a problem that I know the answer to.

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