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After a lifetime of approaching math the wrong way, I took two college math courses this quarter with a newfound zest for math. These classes are integral calc and multivariable calc.

Integral calc started out okay, learning about Riemann sums and the Fundamental Theorem of calculus. But instead of spending a great deal of time gaining the intuition behind these things, we jumped into integration technique after integration technique.

Why on EARTH would we need to memorize and regurgitate a bunch of integration methods on toy problems for 6 weeks? It's absolutely bizarre. I'm not taking this at some random JC either, this is at a top research university. This class is single-handedly destroying my enthusiasm for calculus.

When we have software that can do much more difficult integrals than we can with pencil and paper, why would we waste time memorizing Trig substitutions or integration by partial sums? Doesn't that just make it a glorified algebra class?

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    $\begingroup$ Why on EARTH would anyone taking a carpentry course be asked to spend their time sawing wood? Jez! If you're taking a calculus course then you need to know how to apply calculus. You'll find that there is no need to memorise anything if you really understand what is going on. Integration by parts is an attempt to "undo" differentiation of products while integration by substitution is an attempt to "undo" differentiation of composite functions. $\endgroup$
    – Fly by Night
    Aug 31, 2012 at 0:31
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    $\begingroup$ I think math education should start with understanding number systems and basic counting and go from there, focusing only on intuition. No algorithms. No "techniques". I figured at the university level this dream would be realized. Why are they teaching me so many techniques if I am never going to use them? If you are going to teach me something with no practical application whatsoever, have it be intellectually thrilling. Let me understand the intuition and build a robust mental model of math not built on flimsy "methods" and "techniques" that I don't understand the basis of. $\endgroup$
    – mr real lyfe
    Aug 31, 2012 at 0:40
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    $\begingroup$ @FlybyNight How does sawing wood with a hand saw help you understand carpentry better than using a power tool? $\endgroup$
    – Code-Guru
    Aug 31, 2012 at 1:39
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    $\begingroup$ I don't think integration techniques are just 'techniques'. There are very deep things going on behind. For instance the theory of differential forms and deRham/ Hodge theory are closely related to integration by parts. Actually much modern mathematics is 'generalized' version of those naive looking techniques. So I guess there is some point in familiarizing oneself with them. Another aspect is that one should not think 'too much' about general theory but 'too little' on special examples. Most of the time a good example is sufficient to provide all the insight needed for some grand theory. $\endgroup$
    – Hui Yu
    Aug 31, 2012 at 1:47
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    $\begingroup$ As a side note: just this week I've had to do probably a half-dozen integrals that Mathematica was no help on, and a few others for which I'm still not convinced it gave me the right answer. So I can definitely attest to the value of knowing how to integrate things manually. $\endgroup$
    – David Z
    Sep 2, 2012 at 10:17

18 Answers 18

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This is, in my opinion, a common feeling after a "lifetime of approaching math the wrong way". People are taught math in a very rigid rule-based formula/pattern method, and then when they contrast this against mathematical proofs they have a knee-jerk reaction against anything which looks even remotely like what they did before. The fact of the matter is, however, that you will need to be able to do some of this without aid of a computer.

When reading a proof, it is easy to take for granted that you are able to fill in the details between steps of the proof, when really all these steps are able to be filled in precisely because you have the understanding of solving equations (basic algebra) and working with inequalities (basic arithmetic), for example. The same thing holds true for proofs involving integration and derivatives.

Perhaps it's best to leave it to those who really know what they're talking about - Spivak writes in his chapter on integration that our motivation should be that:

  1. Integration is a standard topic in calculus, and everyone should know about it.
  2. Every once in a while you might actually need to evaluate an integral, under conditions which do not allow you to consult any of the standard integral tables.
  3. The most useful "methods" of integration are actually very important theorems (that apply to all functions, not just elementary ones).

He emphasizes that the last reason is the most crucial.

I would personally advocate that students should be wary of falling into the trap of thinking that such pedantic methods are beneath them. It is often easy to think you understand something at a high level, but you don't truly learn what it is all about until you really get your hands dirty with it.

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    $\begingroup$ 1. Using a quill pen is a standard topic in writing, and everyone should know about it. 2. Every once in a while you might actually need to write with a quill pen, under conditions which do not allow you to use a ball point, fountain, or felt tip pen. 3. Quill pens are based on very important theoretical ideas such as capillary action. $\endgroup$
    – user13618
    Aug 31, 2012 at 5:15
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    $\begingroup$ @BenCrowell that assumes that integration techniques have some other near-substitute that performs better. What would such a thing be, and why does it mean calculus students can dispense with knowing how to integrate? $\endgroup$
    – Robert Mastragostino
    Aug 31, 2012 at 5:39
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    $\begingroup$ No one is saying that calculus students should dispense with knowing how to integrate. That is different from being extremely practiced in hand integration, at the expense of intuition. Take computer programming, for example: If I were to teach someone programming, from the ground up (not that this is the best way to learn programming), I would probably start with assembly (Riemann Sums/FT of calc). Once they had a solid understanding of how the bits are moving around, I may briefly cover C/C++ (integration tricks/techniques). After that, I'd teach them a high level language, like Python. $\endgroup$
    – mr real lyfe
    Aug 31, 2012 at 11:16
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    $\begingroup$ But how on earth do you form an intuition in the first place without doing the work? Solid intuitive understanding is built by working through problems. This may not be obvious now - it came as a surprise to me at undergrad level, since I'd grasped many high-school topics so quickly I never noticed the process of forming an intuition at all - but it's still true. $\endgroup$
    – Useless
    Aug 31, 2012 at 11:31
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    $\begingroup$ On your programming analogy: your teaching as described will exactly be memorizing facts (about binary, assembly and C) and will never form the solid understanding you describe, unless you make them actually code real projects and fix real bugs. $\endgroup$
    – Useless
    Aug 31, 2012 at 11:35
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A short answer from Isaac Asimov : 'The Feeling of Power'.

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    $\begingroup$ Nine times seven, thought Shuman with deep satisfaction, is sixty-three, and I don't need a computer to tell me so. The computer is in my own head. $\endgroup$
    – Nanashi No Gombe
    Feb 9, 2018 at 10:35
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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.

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    $\begingroup$ Haha now I understand... I had "integration by parts" when I was in pre-university. When they did it all over again in university, students were unhappy. But honestly, even having heard and practiced, doesn't mean we could still remember how to do it! $\endgroup$
    – Jake
    Aug 31, 2012 at 5:17
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    $\begingroup$ This is sensible for engineering majors, but here in California, biology majors at UC take the same freshman calc course as engineering majors, because it's a coreq for the physics they're required to take. For them, calculus is essentially a terminal course. Any technique they learn in the last week of Calc II can only be applied in the last week of the physics course, i.e., really not at all. I teach these students at a community college. They're going to be physical therapists, dentists, and pharmacists. The situation you're describing is never going to happen to them. $\endgroup$
    – user13618
    Sep 1, 2012 at 14:22
  • $\begingroup$ Ugh I hate how relatable this is now that I'm dealing with some interdepartmental politics. $\endgroup$
    – Cameron Williams
    Nov 25, 2021 at 16:26
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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.

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    $\begingroup$ In my opinion, saying that learning integration techniques involves no critical thinking is ridiculous. Even in the extreme situation that students are only learning methods and none of the thinking to get to them, you are still excluding the process of deciding the best method to use, which in many problems is nontrivial. If your complaint is that sometimes classes just teach the methods with no nontrivial examples, then it sounds like something is wrong with those classes, not the practice in general. $\endgroup$
    – Sean Clark
    Aug 31, 2012 at 5:09
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    $\begingroup$ @user1306: "Even in the extreme situation that students are only learning methods and none of the thinking..." This is not extreme, it's nearly universal. How often have you interacted with students who have completed a year of calculus and are then asked to apply their skills? I teach physics for a living, and I interact with these students every day of my working life. Almost none can tell whether an odometer differentiates velocity or integrates it; manipulate a function stated in terms of letters other than x and y; or handle a function written with symbolic constants. $\endgroup$
    – user13618
    Aug 31, 2012 at 5:27
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    $\begingroup$ I understand your point, but do not completely agree for two reasons. 1: Someone has to program the computers. 2: Working through a proof without knowing integration by parts will probably be quite hard. Sure, you can let a computer perform the integration, but in that case it will simply be magic to you. It is basically the same with arithmetics. Sure, the computer can do it better/faster than you but does this mean you should skip learning how to do it yourself? $\endgroup$
    – Leo
    Aug 31, 2012 at 11:04
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    $\begingroup$ @ordinary Most people do not have a truly fundamental comprehension of arithmetic. Most people have just memorized a set of rules which they were taught in order to get the right answer, and they only know it's the right answer because they are told it is the right answer. $\endgroup$
    – mboratko
    Aug 31, 2012 at 23:33
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    $\begingroup$ @Ben I can not agree that calculating few hard integrals during the process of education is pointless. If someone has never had to drive a nail he will not appreciate the hammer. So one reason to learn integration techniques is to see how hard it is, you will not appreciate numerical analysis, Monte Carlo methods and much of "applied mathematics" without experiencing those difficulities yourself. Using your terminology I think that even R's will benefit from doing some "stupid" and "mechanical" calculations. Of course we should keep in mind that introductory Calculus is not about computing $\endgroup$
    – Godot
    Sep 1, 2012 at 0:26
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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...

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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 $\displaystyle\int_0^1 xe^x dx$ (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.

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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 relationship between differentiation and integration. It would be a shame to not see this duality carried as far as it 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.

Pursuing duality whenever and wherever you find it is a good habit to form as you become a mathematician. Integration techniques are perhaps the first baby steps to developing this habit. However, this is perhaps the most superficial reason to appreciate integration techniques though.

  1. Integration techniques are computationally useful for proofs.

More so than the exercises would have a student believe. It's true that most of the beginning problems in a calculus text are rather repetitive and tedious. These problems however are sort of like training wheels.

One of the best ways to exemplify that integration techniques are useful is to explore recurrence relations. These types of problems are usually some of the latter exercises in calculus texts. 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 $$\frac{2}{1}\cdot\frac{2}{3}\cdot\frac{4}{3}\cdot\frac{4}{5}\cdot\frac{6}{5}\cdot\frac{6}{7}\cdots=\frac{\pi}{2}$$ which is quite beautiful in itself, relating $\pi$ to the odd and even positive integers.

It should also be mentioned that one of the earliest and most common proofs that $\pi$ is irrational invokes integration by parts.

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.

  1. 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 is a quote that is relevant here. Namely,

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

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 a powerhouse of a tool at our disposal it is natural to go on and use it to calculate the things we wonder about. Some of the problems that you encounter in calculus texts are sometimes actually problems that the author came across naturally themself or found novel enough to include.

With elementary 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. Finding the limits of your shiny new toy is also a good habit to form as you become a mathematician. The integration techniques you learn in a standard calculus course is a roadmap through the genius of past mathematicians that you don't have to repeat or 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 $$e^{-x^2}\qquad \frac{\sin x}{x}\qquad\frac{1}{\ln x}$$ 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.

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  • $\begingroup$ I honestly don't know why this answer doesn't have more upvotes... $\endgroup$
    – Dude156
    Apr 15, 2019 at 0:58
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    $\begingroup$ @Dude156 old questions with a lot of votes and already great answers don't get as much traffic. A few users still find value in answering them when they see something missing for future users. $\endgroup$
    – user123641
    Apr 17, 2019 at 11:20
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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.

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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.

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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?

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    $\begingroup$ This is a false analogy. Talking is an important ability in everyone's daily life. Doing hard integrals isn't. We will always have a class of experts who know how to do hard integrals by hand, just as we will always have people who know how to use a slide rule and log tables. The question is how big that class of experts should be. It should be shrinking, just as it was appropriate for the class of expert slide-rule users to shrink. But instead we're making it grow to include, for example, community college students who want to be physical therapists. $\endgroup$
    – user13618
    Sep 1, 2012 at 17:29
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    $\begingroup$ @BenCrowell: I disagree with your opinion about taking integrals and the ideal number of experts. Taking integrals, reading, talking, writing poetry or any expression of human knowledge should be an important ability in everyone's life. Anyway, the analogy is correct because OP's condition for "unnecessary" personal knowledge was if could be applied by a computer. But as yours, this is an opinion. $\endgroup$
    – dot dot
    Sep 2, 2012 at 10:55
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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.

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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.

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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.

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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?

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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.

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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?

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  • $\begingroup$ I removed your third point since it did not answer the question and violated this site's rules about civility. $\endgroup$
    – N. F. Taussig
    Mar 22, 2015 at 11:59
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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.

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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.

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