I dare to ask a question similar to a closed one but more precise.
Are there any established textbooks or other serious published work that use $\int^x$ notation instead of $\int$ for the so-called "indefinite integrals"?
(I believe I've seen it already somewhere, probably in the Internet, but I cannot find it now.)
So, I am looking for texts where the indefinite integral of $\cos$ would be written something like: $$ \int^x\cos(t)dt =\sin(x) - C $$ or $$ \int^x\cos(x)dx =\sin(x) + C. $$
(This notation looks more sensible and consistent with the one for definite integrals than the common one with bare $\int$.)
IMO, the indefinite integral of $f$ on a given interval $I$ of definition of $f$ should not be defined as the set of antiderivatives of $f$ on $I$ but as the set of all functions $F$ of the form $$ F(x) =\int_a^x f(t)dt + C,\qquad x\in I, $$ with $a\in I$ and $C$ a constant (or as a certain indefinite particular function of such form). In other words, I think that indefinite integrals should be defined in terms of definite integrals and not in terms of antiderivatives. (After all, the integral sign historically stood for a sum.)
In this case, the fact that the indefinite integral of a continuous function $f$ on an interval $I$ coincides with the set of antiderivatives of $f$ on $I$ is the contents of the first and the second fundamental theorems of calculus:
the first fundamental theorem of calculus says that every representative of the indefinite integral of $f$ on $I$ is an antiderivative of $f$ on $I$, and
the second fundamental theorem of calculus says that every antiderivative of $f$ on $I$ is a representative of the indefinite integral of $f$ on $I$ (it is an easy corollary of the first one together with the mean value theorem).