# Are there mathematicians that dislike integral calculus?

• B
Gold Member
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

PeroK
Homework Helper
Gold Member
2020 Award
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
S.G. Janssens
Are there any mathematicians (at any education level) that hate doing integrals by hand?
Yes, you.
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
fresh_42
Mentor
My math teacher used to say: "Everybody can differentiate. But it takes an artist to integrate."

• • • Einstein44, vela, rudransh verma and 6 others
Staff Emeritus
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
Svein
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
jasonRF
Gold Member
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
mathman
Solving differential equations is worse.

• benorin and jasonRF
FactChecker
Gold Member
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
Ssnow
Gold Member
An integral are like sum, if you are a mathematician how you can hate to sum things?
Ssnow benorin
Homework Helper
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 gonna 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
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
Twigg
Gold Member
Integrals can be plenty exciting! (Sorry, couldn't help myself.)

• martinbn, jasonRF and benorin
symbolipoint
Homework Helper
Gold Member
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
fresh_42
Mentor
"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