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It is the very symbol of "indefinite integral" that is flawed and confusing. It should be removed and kept only as a "guilt practice", like treating $dy/dx$ as a real fraction and things like that.

I came across this statement as a well-received comment on another question. I'm interested in understanding what the reasons are for this position, primarily because I'm curious about why something I've been taught since childhood might be wrong.

Also, as I'm starting to study mathematics at postgraduate level I'm seeing definite integration used more often. For example initial conditions are often applied directly as in $y(0)=0, f(y)\,dy/dx = g(x) \implies \int_0^yf(u)\,du = \int_0^x g(v)dv$, and I'm having to get used to seeing solutions written out with embedded definite integrals where analytic results can't be found. With this comment in mind I'm wondering if a more parsimonious approach would be better, dropping indefinite integrals altogether.

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This seems to be a question about mathematics instruction, rather than about mathematics. –  GEdgar Mar 17 at 15:04
    
@GEdgar The answers are certainly instructing me. Beforehand I thought that indefinite and definite integrals were pretty much the same thing. I was the "clumsy student" of one of the answers :). I am starting to see the point that was made in the comment I quoted. –  TooTone Mar 17 at 15:27
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Why should we get rid of indefinite integration? - We shouldn't. :-) –  Lucian Mar 17 at 16:44

4 Answers 4

Indefinite integration is most often used to denote anti-differentiation, which leads students to believe that integration and anti-differentiation are the same thing, which is absolutely not true. There are plenty of functions which have anti-derivatives but are not integrable, and functions which are integrable but don't have anti-derivatives. The use of anti-derivatives isn't the problem, it's the term "indefinite integral" and the use of an integral symbol for them that's the problem.

Edit: I assume we're talking about Riemann integration here. Take any function $f$ which is differentiable on some interval such that $f'$ is unbounded. Then $f'$ has an antiderivative, namely $f$, but is not integrable on that interval since integrable functions must be bounded. The issue is that students come to believe that the Fundamental Theorem of Calculus tells us how to compute all integrals, when in reality this only applies to certain integrals.

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Can you please explain what you mean by “having antiderivatives and not being integrable”? –  egreg Mar 17 at 13:07
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A classic example of such an $f$ (which, in fact, has a bounded derivative) is the Volterra function –  Omnomnomnom Mar 17 at 13:16
    
@Omnomnomnom, nice. –  Santiago Canez Mar 17 at 13:17
    
@SantiagoCanez Thanks; this of course doesn't happen when continuous functions on compact intervals are considered (which is why asked for a clarification). –  egreg Mar 17 at 13:28
    
I totally agree with this point of view. Indeed, what I meant with the above quote was the introduction and diffusion of a symbol like $D^{-1}$, instead of $\int(\ldots) dx$, to denote antidifferentiation. Of course, a seasoned mathematician can easily use the notation she wants, but for a beginner, the confusion between antidifferentiation and integration is hard to overcome. –  Giuseppe Negro Mar 18 at 18:32

Primitives are useful (Barrow's rule!). I agree that the name "indefinite integral" and the symbol used can be confusing for some (clumsy) students.

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The indefinite integral is simply the set of antiderivatives of a function. Once can prove that all antiderivatives differ by some constant, hence the $+C$. But, one can define the set of antiderivatives without any mention of the concept of integration. It is only after the Fundamental Theorem is discussed that one can see why we use the integral symbol.

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Thankyou for a very clear explanation. I assumed I knew what an indefinite integral meant but I didn't. –  TooTone Mar 17 at 15:25

It is true that, unlike derivatives, exact primitives soon leave room to estimates (i.e. approximate solutions and so on) in many branches of mathematical analysis. If you study differential equations, you'll see that only few equations can be actually solved by computing primitives.

However, I wouldn't stop teaching (indefinite) integration to my students. I agree that we spend too many hours doing exercises on primitives, since no working scientist will ever compute by hand things like $$ \int \frac{x^4-3}{x^5+5x^3-x+1}dx. $$ On the contrary, we all learned that the only reasonable way to compute definite integrals is th apply the Fundamental Theorem of Calculus. Therefore, sooner or later students should learn the calculus of primitives. Talking about notation, would it be better to write $$\int f(x)\, dx = \mathcal{I}(f)?$$ Some books do this, but I tend to believe that instructors like this more that students will ever do.

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Though it's true that computing primitives is of little practical value anymore, after being pushed out both by computer numerical computations and computer symbolic computations, I would be sad if there were no place in any calculus class for learning to do such integrals. They are, after all, fun and mind-expanding. Why does calculus have to be only professional preparation? –  Ryan Reich Mar 17 at 20:55

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