Why should we get rid of indefinite integration?

Question

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.

Answer 1

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

Answer 2

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.

Answer 3

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.

Answer 4

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.