Can, $$\int_{}^{x}f(\mathrm{t}) \mathrm{dt}$$

be considered as another representation of an indefinite integral?

  • $\begingroup$ I'm hesitantly saying yes. I don't see a problem with it but it doesn't feel right. $\endgroup$ – Jacob Claassen Jun 21 '17 at 5:08
  • $\begingroup$ I could have an explanation. Let us have the integral function $y(x)$ to be continuous and differentiatiable in an interval $I$. The antiderivative of its derivative is, $g(x) =y(x) +C$. If the lower limit were specified as say $a$, the second part of the Fundamental Theorem of Calculus would give us, $y(x)=g(x) - g(a)$, $g$ being the antiderivative of the integrand in $y$. Here in my represent, we know not what $a$ is. Might as well write $y(x) =g(x) - C=g(x) +K$. We know that, an indefinite integral is expressed as a function of $x$ plus a constant. Here, $y(x)$ with no lower limit fits. $\endgroup$ – R004 Jun 21 '17 at 5:23
  • $\begingroup$ $y(x)=\int_{}^{x}f(t)dt$ $\endgroup$ – R004 Jun 21 '17 at 5:24
  • $\begingroup$ so what's the benefit of using this notation? It just adds a new letter and step. $\endgroup$ – Jacob Claassen Jun 21 '17 at 5:26
  • $\begingroup$ I believe it helps me accept the integral notation for the indefinite case. $\endgroup$ – R004 Jun 21 '17 at 5:28

I have a feeling I've seen your notation used elsewhere (the nagging thought at the back of my head is that Russian authors used it, but I haven't got any examples to hand to verify that), though I think the variable $x$ is usually written at the foot of the integral sign rather than the head. This has one (tiny) benefit over your notation: as it stands a hurried reader might think that the lower integral is supposed to be $0$ and its absence is a typo.

Two general rules for (all) notation are:

  1. Be clear and consistent: if you're going to introduce something new explain it when it appears and use it consistently throughout
  2. Don't be redundant. Introducing a new notation for $\cos x$ will annoy and confuse people because there is perfectly good existing notation already. Introducing $f(x) := \sum_na_n\cos (nx) + b_n \sin(nx)$ is useful shorthand and may eventually become standard (Fourier analysis).

Rule 1 has a corollary as well: explain why you've introduced it. If you can't do that satisfactorily then you've probably broken rule 2.

For me your notation doesn't fall foul of rule 2 because there isn't a generally accepted standard for functions that are indefinite integrals of integrable functions -- switching to the capital version letter is used, but is not the only choice. So I would say go ahead and use it, respecting rule 1 above.

  • $\begingroup$ I have tried explaining why I chose this notation in the comments above. I hope it makes sense. $\endgroup$ – R004 Jun 21 '17 at 5:46

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