Jtdylktuy

A short answer from Isaac Asimov : 'The Feeling of Power'.

Here is a little fable. Some parts may resemble actual events that occurred someplace or other. Other parts are purely made up.

A large university offers different types of calculus courses for the benefit of various other departments. One day the Mechanical Engineering curriculum committee comes to the Mathematics curriculum committee.

"Look!", they say. "If we tell prospective students that we require $X$ semester hours of math for our major, but a competing university says they require only $X-2$ hours, that other place will win the students. So we have to cut $2$ hours from the required math courses."

"OK," Math says. "What do you want to cut?"

Mechanical Engineering studies the syllabi, and says (among other things), "Cut out techniques of integration. Students can do all that by computer, anyway."

So the new--sleeker--course for Mechanical Engineers comes to be.

A few years pass.

Big important Senior Professor is teaching a course for Mechanical Engineers. He derives an equation for this problem he is doing. Then he says, "Now we integrate by parts to put the problem in this other form, so we see it means that we should minimize the energy."

The students reply: "Integration by parts? We've never heard of that!" (Actually, integration by parts had been mentioned a couple of times in their math course, but they had no baby problems to practice it on, and thus they have forgotten it.)

NOW Mechanical Engineering is accusing Mathematics of shoddy work... The members of both curriculum committees have changed by then, so neither department is likely to remember the reason that techniques of integration was omitted from the integral calculus course.

Somebody has to play devil's advocate here, so I guess it's up to me.

The second semester of freshman calc is universally devoted to force-feeding students with tricks for integration which they will neither need nor retain after their final exam. This wouldn't be so bad if there were some other educational purpose being served. For comparison, a student taking an English class might neither need nor retain any specific skills or knowledge related to the novel White Teeth by Zadie Smith, but the student is arguably developing skills such as critical thinking and written expression. There is unfortunately no such argument to be made for techniques of integration. Students learn this bag of tricks as a set of cookbook procedures -- the opposite of critical thinking.

The analog of the skill in written expression taught in an Engligh course would be facility in performing integration. This analogy fails, however, because written expression is a skill that can only be performed by the human mind, whereas computers can now carry out integration so well that the human who can outperform Wolfram Alpha in integration is as rare as the human who can (on a good day) win a game of chess against Deep Blue. Since computers don't seem to have attained creativity or self-awareness, it should be deeply demoralizing to the practitioners of any art when their craft is computerized. It shows that what they were doing could have been done better by a mindless automaton. For this reason, it's unlikely that humans a hundred years from now will be much interested in chess or integration techniques.

In the nineteenth century, men who wanted to become officers in the British military were required to demonstrate proficiency in ancient Greek. The intention was to exclude the working class. Today, people who want to be doctors are required to demonstrate proficiency in techniques of integration. The intention is to weed out students from the "impacted major" of biology at the University of California. In defense of the nineteenth-century British ruling class, we can observe that translating English into ancient Greek is still a skill that can't be carried out by a computer.

It is good to know how to integrate functions even for those three reasons:

1. You do not have to turn on your computer to check every triviality.




2. You can check if a black-box program is not giving you total bollocks.




3. You can amaze people by actually showing how to get formulas for cones' volumes, bottle volumes etc...

I would argue that the purpose of these techniques (to a mathematician) is not so much the value of integrals but how the integration techniques enrich advanced mathematics.

Typically the techniques studied are integration by parts, trig substitution, and partial fractions. It would be possible to enumerate countless uses of these ideas in the literature, but I will restrict myself to only a few.

Integration by parts shows up all over the place in mathematics, especially in differential equations, which will be an important course for anybody studying math. The first place you'll probably run into it there is doing Laplace transforms, and especially when proving properties of the very important gamma function.

If you happen to go on farther into partial differential equation theory, you'll study Sobolev spaces which are roughly spaces of functions who have partial derivatives, but they are actually weak derivatives which are defined in terms of integration by parts. Sobolev theory is exceptionally abstract and it would be a huge pain to have to introduce integration by parts before studying them.

Partial fraction decomposition will also be used in differential equations to find inverse Laplace transforms. It's not the only way of doing this, but you'll see them there for sure. Again, it would not be advisable to introduce partial fractions here, since inverse Laplace transforms are already difficult enough deep into a DE's course.

Trigonometric substitution forms a good base for studying elliptic integrals, especially Jacobi elliptic functions.

So really, the point isn't to make it so you can compute $displaystyleint_0^1 xe^x dx$ (though you should be able to or you look silly!), but rather so you are able to comprehend more advanced and abstract things that use these techniques as part of their definition or part of their theory in some way.

Why do we need to learn integration techniques?

Because it is very useful and is the foundation for higher mathematics.

Personally, I never understood the objective of maths lessons until I started solving real world problem in my working life. I realised that one important lesson to be brought home from these "boring" lessons is not how to do integration or calculus itself, but rather, to be able to spot integration and calculus problems in real life and then formulate a mathematical statement about the problem and then use a computer to give you results.

I am currently a software engineer. I use calculus for programming graphics and animations; cost optimisation for manufacturing; and provide quantitative financial analysis on graphs and charts. There may be various methods and conceptual approach to these problems, but realising calculus is also one of them can help a lot.

Perhaps you might like this TED talk.

http://www.youtube.com/watch?v=60OVlfAUPJg

It is a very interesting question and I am not exactly sure where to draw the line, but it has become very clear that learning to use a high-end CAS will have amazing benefits in the learning experience, IMO.

Applying the same reasoning, why do you talk if a computer also can do it? In one word: independence. Computers run algorithms created by humans, but, what if one day the computer doesn't know an answer to your integral? What if it's badly programmed? What if the computer breaks? Mathematical problems in life would still there. Who would solve future problems? What would you do?

I believe knowing different techniques does provide you with familiarity with commonly encountered expressions and techniques in other situations. For example, the partial fraction decomposition is important whenever you see a linear transformation. Integration is just one of many linear transformations. Green's identity (integration by parts) is also super-important, and there are several occasions your math software cannot help you with this. The less important topic could be the half-angle tangent substitution, but it's applicable in so many cases, so why don't we just memorize this one technique, right?

Anyway, I agree that 8 weeks of just integration may not be a good thing. If you don't see how the techniques are used, you may not realize their importance, and forget them eventually. That happens a lot. Maybe the course is organized like this for the convenience of the teachers and TAs(?) Just my 2 cents.

It seems to me the initial qualm is with the lack of understanding provided with practice. If you do not understand why 2 + 2 = 4, why practice it? Just use a calculator. Practice only becomes effective when you know the reasoning behind what you are doing and why the method works. A lot of times, in math courses, you are given tips and tricks to practice without ever being taught how they work or how to look for situations where they become useful. It is correct to say that you do not need to memorize things once you have a true understanding. Why introduce topics to memorize and not provide a true understanding? This seems to be the OP's issue.

The problem is that once you hit Ordinary Differential Equations a lot of those turn out to be needed, in fact the specific ones that are included is almost certainly based on "ones we will need later". In a very difficult course for students like ODEs if you take time to introduce a new integration method (because you didn't already cover it) you will make the course impossible for the vast bulk of students because there would be too much to take in.

Some integration techniques will come up in one form or another again and again (substitution and integration by parts) in other courses and need to be as natural as multiplication and division for advanced mathematics. The more specialized ones you mention (trig substitution and partial fractions) also play the role of forcing you to do enough practice with using the fundamental ones as part of a more complicated problem to ensure you really get serious practice with them, i.e. can use them in a context bigger than a single problem where that is the only thing you have to do.

I think there are some students like me who want to study math in way that each statement is examined like how it got here what axioms/hypthesis caused it to be true, what if an axioms is removed then how it will effect on that statement. Instead of giving ready-made theorems, I would rather try to prove in timely fashion with some intermediate theorems as hints. Many of us want to (re-)discover it, rather than learn it.

So when these integration techniques are presented and the students hold-off, it just takes the fun away, not because you can't do it in your free time but because the free time is way shorter than the time needed to go thoroughly into such vast subject. so the choices you are left with is either curb your enthusiasm and pass the exams with the question "Do I really understand it?" or fail the exams with the smile on your face and satisfaction saying "That felt good".

The point is to devide a subject more flexibly so that no one feels force-feeded. So what if someone doesn't feel that something is not important, force feeding won't benefit him or the society. Instead letting people do what they want to do would likely help the advancement in their life and the society itself.

There would be people who want use math solving program and there would be ones who want to create it. The point is (again), that people from both side will exist. Even if one is in majority. so what @ordinary asked does he have to learn the intergration methods even if the same can be done by a software or the methods is not going to be needed in future? Even he belongs to the minority (the math-rediscoverer), he does not. Maybe he will realize otherwise in future or may be he won't? If he does, there is nothing wrong with going back and read/learn it but this time with motivation. If he does not, time well saved. So I think it is best to leave it to the future. No point in force feeding now and even if people like @ordinary become majority (may be they are), it does not mean the other side will go extinct?

Why integrals? Why not computers? Why learn?

1: It's the framework to understand many "advanced" techniques and technology topics used in different areas of the science.

• This means you will be a professional on what you are doing. The one who is not just using technology but knows how it works or even can reproduce the technology.


• Trivial things will be understood.


• Someone has got to know about it.

2: How would you talk to the computer when needed if you don't even know the function names?

Integrals and integration are parts of mathematics. Mathematics is a language that translates abstract ideas into formulas, definitions and equations. The better one knows the elements of the language of mathematics the easier is to understand mathematical proofs and books.

This is a fairly old question, but I didn't see an answer that reflected my point of view. So here's my two cents. There are three primary reasons (other than the ones that have been belabored already) that I believe learning integration techniques are worth-while:

1. Learning integration techniques reinforces the importance of duality.

The Fundamental Theorems are beautiful in part because they encapsulate the inverse relationships between differentiation and integration. It would be a shame to not see these inverse relations carried as far as they could be. For example, we might expect that every property that differentiation satisfies, there might be an inverse relationship that integration satisfies. And, indeed, integration by parts can be thought of as the "opposite" of the product rule. And change of variables can be thought of as the "opposite" of the chain rule. This is the most superficial reason to appreciate integration techniques though.

2. Integration techniques are actually computationally useful for proofs.

More so than the exercises would have a student believe. Some of the best ways to exemplify this is the discovery of recurrence relations. For example, if we defined
$$I_n=int_0^pi sin^n x, dx$$
for every natural number $n$, integration by parts would imply
$$I_n=frac{n-1}{n}I(n-2),.$$
This small observation formed the basis of John Wallis's proof of the identity
$$frac{2}{1}cdotfrac{2}{3}cdotfrac{4}{3}cdotfrac{4}{5}cdotfrac{6}{5}cdotfrac{6}{7}cdots=frac{pi}{2}$$
which is quite beautiful in itself, relating $pi$ to the odd and even positive integers.

A similar technique with the logarithm could should that
$$1-frac{1}{2}+frac{1}{3}-frac{1}{4}+frac{1}{5}-cdots =ln 2$$
which is also aesthetically interesting.

3. Leaning integration techniques informs you about what is possible and hints at what is impossible. These hints of impossibilities lead to some beautiful mathematics.

There are two quotes that are relevant here. Namely,

"Common integration is only the memory of differentiation..." - Augustus de Morgan

"It can be of no practical use to know that $pi$ is irrational, but if we can know, it surely would be intolerable not to know." - Edward Charles Titschmarsh

Integration is interesting precisely because it is difficult. Without the FTC, calculating integrals with Riemann (or Upper/Lower) sums is tedious and frequently requires a spark ingenuity which changes dramatically between the functions you want to integrate. With the FTC however, many elementary functions become quite easy to integrate. With these functions mastered, it is natural to move on to more complicated functions to probe the limits of the FTC, integration by parts, change of variables, etc. to see where we eventually hit a wall. The techniques you learn is a roadmap through the genius of past mathematicians that you don't have to repeat, but would have difficulty redeveloping on your own.

Despite the swath of functions you're able to integrate with these techniques, it is interesting that there is a wall. Namely, we cannot integrate
$$e^{-x^2}qquad frac{sin x}{x}qquadfrac{1}{ln x}$$
using the techniques that we learn from Calculus. We would not have been able to find these functions without getting our hands dirty through extended and purposeful effort. Few courses take integration to this extreme, much less prove that we cannot integrate these functions in terms of our other elementary functions.

It should also be mentioned that integration by parts also leads to an elementary proof of the irrationality of $pi$...

OK. This is actually opposite from my notion of doing integrals, and so it's like I'm to say this.
The thing is, yeah , you're right sometimes there are instances when you have to take the substitution or technique being used at face value, just because it's working... You can go to great depths but it may or may not do any good. The essential thing which you are doing by solving n number of questions is that you're developing an intuition for it, for when to use that technique, or what cocktail of substitutions will reduce it to a solvable one. You never never memorise the integrals, except for base cases, and that too for convenience. (Though if you insist, you can indeed go to the fundamentals to derive them even) .
Calculus is essentially the boundary between applicability and theoretical aspects of maths. You're not gonna use things like limit of sum for every damn problem you encounter, right? Also, since you talk of computers, they're just gonna use numerical approx. To give you the value but they're just numbers! You won't make sense of 5.56832799683...
unless you know it's pi^1.5 . Also you don't get the intuition of what and why this integral has this value (**physicist's viewpoint **).
And believe me it's more like a puzzle: every integral is like a battle, and you have to find a battle plan specially for that, every integral needs a custom technique. Rarely it's a question why this worked. So you're just developing that skill set, or in words of Feynman, the Toolbox necessary to take them down. Sure you can use softwares, but where's the fun in it!

PS: Sad you had bad experience in an integral class, but integration is one of the best things one can enjoy, atleast much more interesting than writing down a proof..

Search for a book by the name of "Irresistible Integrals" , I'm sure your notion about integrals will take a roundabout.

we all need integration, and it also help us to how to calculate and to know how to solve problems in mathematics in anyway where we find ourselves.
example:how to differentiate inverse and definite integration and also indefinite integration, so i am with suggestion that all mathematics students need integration.