Textbooks that use notation with explicit argument variable in the upper bound $\int^x$ for "indefinite integrals."

Question

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 $\int$.)

Some context.

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:

1. 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

2. 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).

Answer 1

That notation is used in the classic textbook Elementary Differential Equations by William E. Boyce and Richard C. DiPrima, at least in the third edition (1976), which is the one I have. Quoting from p. 11 (beginning of Chapter 2):

     The simplest type of first order differential equation occurs when $f$ depends
only on $x$. In this case $$y'=f(x)\tag2$$ and we seek a function $y=\phi(x)$ whose derivative is the given function $f$. From
elementary calculus we know that $\phi$ is an antiderivative of $f$, and we write $$y=\phi(x)=\int^xf(t)dt+c,\tag3$$ where $c$ is an arbitrary constant. For example, if $$y'=\sin2x,$$ then $$y=\phi(x)=-\frac12\cos2x+c.$$      In Eq. $(3)$ and elsewhere in this book we use the notation $\int^xf(t)dt$ to denote
an antiderivative of the function $f$; that is, $F(x)=\int^xf(t)dt$ designates some
particular representative of the class of functions whose derivatives are equal to $f$.
All members of this class are included in the expression $F(x)+c$, where $c$ is an
arbitrary constant.

P.S. On second thought, I'm not sure Boyce & DiPrima use the notation $\int^xf(t)dt$ in quite the same way you do. For them the general solution of the differential equation $$y'=f(x)$$ is $$y=\int^xf(t)dt+c$$ since $y=\int^xf(t)dt$ is some (unspecified) particular solution; but for you I think $$y=\int^xf(t)dt$$ is already the general solution.