Textbooks that use notation with explicit argument variable in the upper bound ∫ x for "indefinite integrals."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 ∫ x notation instead of ∫ 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: ∫ x cos ⁡ ( t ) d t = sin ⁡ ( x ) − Cor ∫ x cos ⁡ ( x ) d x = sin ⁡ ( x ) + C .(This notation looks more sensible and consistent with the one for definite integrals than the common one with bare ∫.)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 ) = ∫ a x f ( t ) d t + C , x ∈ I ,,with a ∈ 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, and2. 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).

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Textbooks that use notation with explicit argument variable in the upper bound       ∫    x   for &quot;indefinite integrals.&quot;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       ∫    x   notation instead of   ∫ for the so-called &quot;indefinite integrals&quot;?(I believe I&#039;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:      ∫    x    cos  ⁡  (  t  )  d  t  =  sin  ⁡  (  x  )  −  Cor       ∫    x    cos  ⁡  (  x  )  d  x  =  sin  ⁡  (  x  )  +  C  .(This notation looks more sensible and consistent with the one for definite integrals than the common one with bare   ∫.)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  )  =      ∫    a    x    f  (  t  )  d  t  +  C  ,    x  ∈  I  ,,with   a  ∈  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, and2. 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).

Regelspur8xgdl · Accepted Answer

Step 1That 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 dependsonly on x. In this case                     (2)                              y          ′                =        f        (        x        )             and we seek a function   y  =  ϕ  (  x  ) whose derivative is the given function f. From elementary calculus we know that   ϕ is an antiderivative of f, and we write                     (3)                    y        =        ϕ        (        x        )        =                  ∫          x                f        (        t        )        d        t        +        c        ,            where c is an arbitrary constant. For example, if       y    ′    =  sin  ⁡  2  x  ,, then   y  =  ϕ  (  x  )  =  −      1    2    cos  ⁡  2  x  +  c  ..In Eq. (3) and elsewhere in this book we use the notation       ∫    x    f  (  t  )  d  t to denote an antiderivative of the function f; that is,   F  (  x  )  =      ∫    x    f  (  t  )  d  t 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.Step 2P.S. On second thought, I&#039;m not sure Boyce and DiPrima use the notation       ∫    x    f  (  t  )  d  t in quite the same way you do. For them the general solution of the differential equation       y    ′    =  f  (  x  ) is   y  =      ∫    x    f  (  t  )  d  t  +  c since   y  =      ∫    x    f  (  t  )  d  t is some (unspecified) particular solution; but for you I think   y  =      ∫    x    f  (  t  )  d  t is already the general solution.

Textbooks that use notation with explicit argument variable in the upper bound ∫<!-

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