Another notation for the indefinite integral Can, $$\int_{}^{x}f(\mathrm{t}) \mathrm{dt}$$
be considered as another representation of an indefinite integral? 
 A: 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:


*

*Be clear and consistent: if you're going to introduce something new explain it when it appears and use it consistently throughout

*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.
