# Are there mathematicians that dislike integral calculus?

• B
• docnet
In summary: MIT integration bees and my calculus professor? In summary, while it may not be necessary for mathematicians to be able to solve difficult integrals by hand in a competitive setting, it is important for them to have a solid understanding of basic integration techniques and the ability to recognize common mathematical structures. However, as studies progress towards real analysis, the focus should shift more towards understanding rather than explicit solving. While complex integration and Lebesgue integration may be exceptions, it is perfectly acceptable to use tools like Mathematica for closed form integration. In addition, knowledge of numerical methods and analytical approximation techniques is useful for applied mathematicians. While some mathematicians may dislike doing integr
docnet
Gold Member
TL;DR Summary
Integral calculus bores and frustrates me to nearly to sleep, but differential calculus is, to me, much more interesting and fun. Are there any mathematicians (at any education level) that hate doing integrals by hand?
Solving integrals by hand is difficult and prone to errors, and the techniques such as integration by parts, partial fraction decomposition, and trig substitutions only work for a small subset of integrals and I do not see the point of avoiding technology like Wolfram Mathematica for mathematical research. Is it important for mathematicians in general to have the ability to solve difficult integrals by hand, under pressure, like the students in the MIT integration bees and my calculus professor?

(I have a BA in mathematics and am starting a master's in applied mathematics this fall.)

benorin and Delta2
Perhaps dislike is too strong a word, but it is frustrating that physical problems involving everything from line integrals to the monster integrals of QFT are so difficult to calculate.

Laplace is reputed to have said something like "nature laughs at the difficulties of integration". And he hadn't seen QFT!

Einstein44, benorin, docnet and 1 other person
docnet said:
Are there any mathematicians (at any education level) that hate doing integrals by hand?
Yes, you.
docnet said:
Is it important for mathematicians in general to have the ability to solve difficult integrals by hand, under pressure, like the students in the MIT integration bees and my calculus professor?
In a competitive setting, no. I have always associated integration sorcery more with physicists than with mathematicians. With that being said, of course a mathematician should know the basic integration techniques and the formula manipulation involved, but as your make progress with your studies and move towards real analysis, it should be more about recognizing common mathematical structures than about explicitly solving integrals.

Perhaps complex integration and the associated residue calculus is an exception. It is both a very powerful integration technique (also for integrals of real-valued functions over real domains) as well as a key to important results in spectral theory (a part of functional analysis).

benorin, dextercioby, sysprog and 5 others
My math teacher used to say: "Everybody can differentiate. But it takes an artist to integrate."

Einstein44, vela, rudransh verma and 6 others
docnet said:
Are there any mathematicians (at any education level) that hate doing integrals by hand?
There are mathematicians who never even do integrals.

weirdoguy, docnet and jedishrfu
For physicists: Differentiate is easy, but if you try to differentiate a set of measurements you will end up with just noise (differentiation is a high-pass filter). Integration may be hard, but it tends to remove noise (integration is a low-pass filter).

benorin, S.G. Janssens and docnet
In my opinion there is nothing wrong with using Mathematica to do closed form integration for you. But when it fails, or produces results as something like generalized hypergeometric functions that aren’t always useful, an applied mathematician should have some other tools at hand. I would think numerical methods and various analytical approximation techniques (such as asymptotic expansions) would be good to know. I certainly use them in my work as an engineer.

Einstein44, Twigg, benorin and 4 others
Solving differential equations is worse.

benorin and jasonRF
Mathematicians know the common sorts of integration tricks, a dozen or so, enough so that they can teach them. Those tricks show up enough to be worth remembering. After that, there are books full of integrations where they can look up the integrals that are much harder. Then there are computer tools. But there is much beyond integration that consumes them.
A significant exception which @S.G. Janssens mentioned is complex integration, which has very profound consequences.
Another theoretical exception is the technique of Lebesgue integration, which has great significance.

benorin, S.G. Janssens, sysprog and 1 other person
First year graduate student in math: "Can you direct me to someone who can help me with elliptic integrals?"
Department chair: "We try to get rid of those."

benorin, dextercioby, docnet and 1 other person
An integral are like sum, if you are a mathematician how you can hate to sum things?
Ssnow

I personally love integrals and integration tricks (and of course some special functions oft defined as integrals like the Gamma and Beta functions).

Whereas differentiation is typically procedural: just follow the rules and you get the correct answer (which is not without some degree of fun, but it's easy--I just know somebody's going to post a counterexample here to prove me wrong: they exist). I like integration because it requires skill and ingenuity, and there's always something more to learn.

@fresh_42 often introduces me to new tricks of integration (thanks fresh_42 btw). As you learn more tools for integration it grows on you and can even become fun, I hope. Somebody needs to code the mathematical software that does our integrals for us after all

Last edited:
rudransh verma and fresh_42
docnet said:
Summary:: Integral calculus bores and frustrates me to nearly to sleep, but differential calculus is, to me, much more interesting and fun. Are there any mathematicians (at any education level) that hate doing integrals by hand?

Solving integrals by hand is difficult and prone to errors, and the techniques such as integration by parts, partial fraction decomposition, and trig substitutions only work for a small subset of integrals and I do not see the point of avoiding technology like Wolfram Mathematica for mathematical research. Is it important for mathematicians in general to have the ability to solve difficult integrals by hand, under pressure, like the students in the MIT integration bees and my calculus professor?

(I have a BA in mathematics and am starting a master's in applied mathematics this fall.)
I personally love differential equations too, but that’s largely because I like seeing the mathematics of nature play out and differential equations are everywhere in that regard.
Integration is difficult, but the ability to see a process forward and backward is, I believe, an important skill to being a good mathematicians. I don’t think you should give up on integration, just find a new reason to like it.

benorin and symbolipoint
Integrals can be plenty exciting!

(Sorry, couldn't help myself.)

martinbn, jasonRF and benorin
I may have said this already, in a different subforum on PF, but the discussion here reminds me of when a young student asked, or said approximately, "why is multiplication of multidigit numbers easy but division is so hard?" I still do not have a good answer, but I agree and understand that student's viewpoint.

benorin
symbolipoint said:
"why is multiplication of multidigit numbers easy but division is so hard?"

That's easy.

For multiplication, you feed an algorithm with two known numbers in order to get a third:
$$a\cdot b= x$$
For division, you have to guess an input in order to get a determined result:
$$a \cdot x = b$$

benorin and symbolipoint

## 1. What is integral calculus?

Integral calculus is a branch of mathematics that deals with the calculation of areas, volumes, and other quantities that can be expressed as the limit of sums. It is used to solve problems involving continuous change and is an important tool in many fields, including physics, engineering, and economics.

## 2. Why would a mathematician dislike integral calculus?

Some mathematicians may dislike integral calculus because it involves complex calculations and can be difficult to understand. Additionally, it may not be as elegant or abstract as other branches of mathematics, which may not appeal to some mathematicians.

## 3. Are there any alternatives to integral calculus?

Yes, there are alternative methods for solving problems that involve continuous change. Some examples include differential calculus, discrete mathematics, and numerical analysis. However, integral calculus is still the most widely used method for solving these types of problems.

## 4. Can integral calculus be applied to real-world problems?

Absolutely. Integral calculus has numerous applications in the real world, such as in physics (for calculating the trajectory of a projectile), economics (for calculating profit and loss), and engineering (for calculating the volume of a tank). It is a powerful tool for solving practical problems.

## 5. Is integral calculus difficult to learn?

Like any branch of mathematics, integral calculus can be challenging to learn. It requires a strong foundation in algebra and trigonometry, as well as a solid understanding of the concepts and principles involved. However, with dedication and practice, it can be mastered by anyone.

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