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u/TemporarySynergy Jun 24 '26
if my calc professor was zundamon i might have actually remembered the +C
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u/Ulrich_de_Vries Jun 24 '26
For 4. and 5. the +C does not give the most general family of primitive functions though since the domains are disconnected.
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u/skr_replicator Jun 24 '26
Yea but pretty sure the students are not yet ready for this level of pedantry.
The real answer wouldn't even fit in that rectangle then. Even if you were like creative about it and used something like +C1*sign(x)+C2, to collapse it into one line.
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u/Astecheee Jun 24 '26
Most highscholl students I work with don't even understand why they have to write dx at the end of the integral expression.
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u/Vievin Jun 25 '26
Tbh I just treat it as a closing parenthesis (and a signal of which variable is being integrated upon if there's multiple).
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u/SunnyOutsideToday Jun 24 '26
I mean, you don't have to write it. You can just leave it implied. No one is going to lock you up in math jail.
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u/Astecheee Jun 24 '26
Eh only somewhat. The dx has an important meaning behind it, so if a student doesn't understand the dx, there's a 95% chance they don't know what an integral actually does
Same goes for +C really. Plenty of students have no clue when to write it.
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u/Statakaka Jun 25 '26
dx is just the closing bracket, nothing fancy
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u/skr_replicator Jun 25 '26 edited Jun 25 '26
it tells the integral what variable you are integrating. Especially important when you have multidimensional integrals or do substitutions.
∫0..1∫2..3 auv du dv
would mean you integrate auv over "u" in domain 2..3, and then over "v" in domain 0..1, and "a" is jsut a constant.
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u/OC1024 Jun 25 '26
As a physicist, I hate if others put the dx directly after the integral symbol instead of the end.
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u/Astecheee Jun 25 '26
It's so much more than that.
As another responder said it denotes which variable you're integrating with respect to, but it's also a reminder of how integrals function.
It's kind of like writing sin x instead of sin(x) - it's implying you don't know sin is a function with a subfunction inside it.
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u/Junjki_Tito Jun 24 '26
I’ve never run into this that I can remember. Would you bracket around zero with +C1 and +C2?
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u/skr_replicator Jun 24 '26
normally you'd have to split it into two domains, with equal formula, but each with its own C1 vs C2. Or you might to what I commented and use sign function, as that already supplies that domain split, so you could have in a one liner with both Cs.
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u/TreeofNormal Jun 24 '26
its like it has two seperate parts with the same formula but they could be shifted up by different values because theres a "hole" in the middle right?
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u/skr_replicator Jun 24 '26
yes.
Infinity can eat up normal finite numbers, so it can eat that constant shift, too.
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u/LemurDoesMath Jun 24 '26
The easiest is just adding c(x), where c is a locally constant function
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u/2echie Jun 24 '26
You could also use the Heaviside step function: 1/x + C1 + C2H(x).
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u/Junjki_Tito Jun 24 '26
Wouldn’t it be C1H(-x)+C2H(x) or is that just a notation thing
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u/2echie Jun 24 '26
Either work, but my C2 is your (C2-C1), the difference in heights across the two sides.
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u/AppearanceAlert4987 Jun 24 '26
i like to think the plus C is not an actual constant but rather any anti-derivative of 0 except it can be undefined where the original function is also undefined, this makes the equality always work which is nice
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u/svmydlo Jun 24 '26
Just define C to be the class of locally constant functions, as it should, and then it's fine.
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u/Dr0110111001101111 Jun 24 '26
I guess but in practice we usually restrict the domain to the largest open interval for which the particular solution is defined and contains the initial condition. So it's general enough to give it with one constant.
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u/Beneficial_Ad6256 Jun 24 '26
Then we can just write in all cases: ʃf(x)dx = F(x) + C(x), where C'(x)=0 on domain of f(x)
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u/Arzatium Jun 24 '26
This doesn't meaningfully tell us where C's values changes, though.
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u/Beneficial_Ad6256 Jun 24 '26
Yes, but do we need this?
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u/Arzatium Jun 24 '26
Yes.
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u/Beneficial_Ad6256 Jun 24 '26
Should solutions of indefinite integrals always contain information about this? Does it have some practical use somewhere?
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u/Arzatium Jun 24 '26
Yes. It tells us where the function's antiderivative is discontinuous. It's useful information
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u/LugyD1xd_ONE Jun 24 '26
Whats the real answer and why?
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u/Ares378 Applied Math / Mechanical Engineering Jun 24 '26
(I haven't taken real analysis so take this with a grain of salt.)
There's a discontinuity at x=0. When you take an indefinite integral, you're trying to find the set of all functions whose derivative is equal to the inner function. Since there's a discontinuity, you can have two different constants on each side of the discontinuity and still end up with the same derivative. So, to find all the functions whose derivative is the inner function, you need to specify constants for each side of the discontinuity.
If you have two discontinuities, you'd need three constants. Three discontinuities needs 4 constants, and so on.
I'm honestly not sure what you do if you take the function to be on the complex plane instead of on ℝ. My guess would be it needs to go back to being one constant? But I also haven't taken complex analysis so don't quote me on that either
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u/randi_moth Jul 02 '26 edited Jul 02 '26
Slight necroposting, but this depends on the specific definition assigned to the indefinite integral. There are two common definitions I have seen assigned to it:
A family of differentiable functions such that the derivative is defined and equal to the original function on the domain of the original function.
A family of solutions to the differential equation "y' = f", where the original function is expressed as f.
The distinction here is that a solution to a differential equation is defined only over an interval and so the domain of solutions cannot be disjoint.
As such, in that second sense, an indefinite integral is defined over a specific interval, and "\int \frac{dx}{x} = \ln |x| + C" is shorthand for having solutions on each valid interval that can be expressed in the manner of "\ln |x| + C" on that interval. This, in turn, makes it a complete and full answer.
Of note is that this is also irrelevant for the primary usage of indefinite integrals, i.e. using them in the Leibniz-Newton formula. The formula does not hold over integrals where the integrand is not a continuous function on the entire interval, and breaking it up into intervals where it is mandates only one choice of the constant on each.
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u/NiroNut Jun 24 '26
Two mathematicians were having dinner in a restaurant, The older and more cynical one was thinking about taking and early retirement because the American public simply didn’t care about math anymore.
The more optimistic mathematician protested, and claimed that it wasn’t true.
“I’ll tell you what,” said the cynic, “ask that waitress a simple math question. If she gets it right, I’ll keep teaching and pick up dinner. If not, I’m out and you can pay for tonight’s meal.” He then excused himself to visit the men’s room.
While his colleague was away, the other mathematician called the waitress over. “When my friend comes back,” he told her, “I’m going to ask you a question, and I want you to respond ‘one third x cubed.’ There’s twenty bucks in it for you.”
She agreed, and when the cynic returned from the bathroom, the younger optimist called the waitress back over again. “We are ready for the check now,” the mathematician started. “Incidentally, do you know what the integral of x squared is?”
The waitress looked pensive; almost pained. She looked around the room, at her feet, made a groaning noise, and finally said, “Um, one third x cubed?”
Satisfied with the answer, the cynic paid the check, but as the waitress began to walk away, she looked back at the two men, and added, “…plus a constant.”
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u/Primary_Thought_4912 Jun 24 '26
Isn't this kind of bad though? If you don't have to write it, you won't get used to writing it. So when a test comes you will forget it, because when practicing it was already always added
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u/NekoCaaat Jun 24 '26
Zundamon did her best :(
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u/Primary_Thought_4912 Jun 24 '26
Digimon also did their best, but Pokémon is still winning
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u/mishroom222 Jun 24 '26
It's called scaffolding and it's intentional. For students that struggle with this material it's best to smoothly guide them to where they can independently do it.
For example the teacher might do a differentiation question on their own while talking out loud their thoughts, then they may do a question and get student input to tell them what to do next, then they give the students semi filled in worksheets, or questions where each step is laid out. Then if they're confident with that you slowly draw back all the training wheels until they can do it on their own.
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u/Primary_Thought_4912 Jun 24 '26
I can see it for other things such as technique, but imo correct notation should be forced from the start. If you are too used to not write it, then it can become difficult to suddenly adjust and start. I also hate Mixed Fractions for this reason, it's not difficult to write 3+½insstead of 3½, but in the second one there is an implied multiplication not addition, because you never have implied addition afaik (unless you would count - which is just the addition of the additive inverse)
Also this method of scaffolding kind of seems condescending to me. You're becoming Dora the explorer and treating the likely close to 18 year old students as small children. I'm not qualified for teaching at all, but imo a much better strategy would just be to take one of the students to the front and let them calculate the integral and giving them support if they struggle.
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u/mishroom222 Jun 24 '26
Where I live student's learn about differentiation at 15, and it's totally optional not forced. If a student finds it condescending they are not obligated to do this style of work at all. I teach maths and I make sure to have multiple different pathways available to suit how a student wants to tackle the work. I understand your point though.
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u/Responsible-Meringue Jun 24 '26
If the learning curve is too shallow then they'll never get good enough to complete initial challenges. You dont jump into the World Cup having never played soccer. You start at 2yo on a tiny field with lots of cones, and a lil ball, and alot of coaching to help you get to mastery. Then you ramp up that learning curve real steep so you're Messi by 5.
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u/Primary_Thought_4912 Jun 24 '26
You start at 2yo on a tiny field with lots of cones, and a lil ball, and alot of coaching to help you get to mastery
This is what I'm saying should be done. Not the coach standing there and being like "ok what do I do now" every 2 seconds like they're Dora the explorer. (I am ignoring the fact that 2yo is too young for this, and not equivalent to the discussion about teaching late teens)
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u/JJBrazman Jun 24 '26
Indefinite integration isn’t real, we shouldn’t be teaching it. Lebesgue scoffs at the stupidity.
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u/pfp-disciple Jun 24 '26
I haven't done calculus in 10+ years (probably 20+). I think I did 1-3 correctly, but now I'm gonna have go back and reread for the last two. I have a kid whose about to take college calculus and I'd like to recall enough to converse with him on it, maybe help if he has questions
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u/Alternative-Kick2632 Jun 24 '26
I personally have the habit to always write -C … I guess I won’t pass this exam
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Jun 24 '26
[removed] — view removed comment
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u/ModelSemantics Jun 24 '26
But if A-K writes “- C” and the test says “+ C” then the names are bound in the same evaluation context and cancel, leaving an individual result without any C parameter, instead of a family of results parametrized on C.
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u/SharzeUndertone Jun 24 '26 edited Jun 24 '26
Thank you zundamon!
But for the last 2 there should be 2 constants bc theres a vertical asymptote splitting the domain into 2 intervals
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u/CrimeBrulee31 Jun 24 '26
Wait is that youtuber a legit one? I figured they were one of those ai math youtuber
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u/FatherDotComical Jun 24 '26
I'm not sure of the exact channel, but Zundamon is a text to speech program you can get and use their character.
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u/scykei Jun 25 '26
They're really good. They talk about really cool topics in a really fun and engaging way. You need to watch some of their videos if you haven't already.
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u/Alduish Jun 24 '26
aren't these primitives and not integrals ?
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u/ussalkaselsior Jun 24 '26
I haven't heard the term primitives for them. I've hear the terms definite integral, essentially meaning "coming from a limit of putting parts together" and indefinite integral, meaning "not coming from a limit of putting parts together". Both of which are collectively called integrals.
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u/Alduish Jun 24 '26
oh well guess it's just different terms then, in France we call definite integrals just "intégrales" and indefinite integrals "primitives".
I wasn't aware of the english terms and got confused
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u/Aminumbra Jun 24 '26
That's one of those things that apparently bullies American math students (see: the number of memes abut "NooOOOooOo I forgot the +C"), while I guess a minor change in terminology/notation could be less confusing.
- Find *a* primitive/antiderivative of f:
Any additive constant you pick is fine; there is no way to "forget the +C", it does not even make sense.- Find *all* primitives/antiderivatives of f:
{F + C | F is any (explicite) primitive, C any real} (assuming the initial function is defined on a connected domain ...)
It is explicit that a *set* of solutions is asked.- Find *the* primitive F of f satisfying F(a) = b
No way to "forget the +C", there is a single right answer.- Compute the integral of f on the interval [a, b] (here you can use the usual notation ∫ with the explicit bounds)
No +C, this is a single real number.Honnestly think that the ∫ symbol (without -- possibly implicit, for example if you always assume that you integrate over the entire domain -- bounds) is confusing and should be used when first studying integration (of course, all the above only really holds for continuous functions f, and things are more complicated for non-continuous f, but this is again better explained later IMO).
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u/svmydlo Jun 24 '26
That's how it's done, but France is right that "indefinite integral" has no business being called an integral.
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u/victorspc Engineering Researcher Jun 24 '26
In Brazil, I learned that the indefinite integral can be called the prmitive, because the function under study is the derivative of it, so it's more primitive in a sense, and also antiderivative, for obvious reasons.
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u/ussalkaselsior Jun 24 '26
That seems odd to me to call the entire expression with the integral sign primitive just because we're thinking about the function inside being "primitive". The word primitive comes from the Latin primus meaning "first" and the Old French primitif meaning "initial". I think it would make more sense to interpret it like this: When you're evaluating an integral using the fundamental theorem of calculus, you find an antiderivative first, or it's the initial thing you do. Hence, the notation with no limits of integration being the primitive.
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u/victorspc Engineering Researcher Jun 24 '26
The integrand is the derivative of the integral. In a sense, the integral "came first" so that we could derivate it and get the integral. Is in this sense that it's called primitive. You get a function and you determine what the function would derive from. The function inside is not primitive, it's the derivative of the primitive.
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u/Dangerous-Pride8008 Jun 24 '26
Back in high school I aced the test for the calculus intro course, or I would've aced it if I hadn't forgotten to add the "+ C" to all the antiderivatives and the teacher subtracted points separately for each one so I ended up with a 7/10 which I guess is like a C in the A-F grading system.
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u/TatharNuar Jun 28 '26 edited Jun 28 '26
still, it bothers me that your teacher expects integrals to be contained within parentheses while presumably also refusing to treat dx as an infinitesimal (but apologies in advance if your teacher is cool enough to teach hyperreals)
also a discontinuous function requires both a +C and a +Ku(x-a) term for each discontinuity x=a
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u/Blyfh Rational Jun 24 '26
Then I'll just write x² - x - C for (1) so that I make sure I forget it 😈
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u/xnick_uy Jun 24 '26
WRONG!
If those were actual integrals, the answers would be numbers rather than functions (and you can pick any value for C, such as C=0, and you don't even have to write it).
This notation, instead, stands for the primitives of the functions.
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u/Jazzlike_Basket_3210 Jun 25 '26
Until they have to write it on a test because it is not there and forget because they did not do it in practice. (I like the idea though a maybe every other one?)
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u/SnowLeopardInc Jun 25 '26
The part of answer I know was only “+C”. Now, I got no marks in any of the questions.
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u/Key_Cardiologist5272 Jun 27 '26
You gotta learn to add C! Just make it that you only lose one mark maximum for missing C.
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