I would like to ask all the math masters, whether there is dx in the integral expression of the definite integral...

The question of whether the differential 'dx' is a necessary component of the definite integral notation is foundational, and the unequivocal answer is yes. In the standard Leibniz notation, ∫_a^b f(x) dx, the 'dx' is not a superficial appendage but an integral part of the symbolic architecture. It serves the critical function of specifying the variable of integration, which is essential for clarity and computation. Without it, the expression ∫_a^b f(x) is incomplete and ambiguous; it fails to indicate with respect to which variable the integration is to be performed. This is particularly crucial in multivariate contexts, but even in single-variable calculus, its omission renders the expression formally incorrect. The notation binds the function f(x) and the differential dx into a single integrand entity, representing an infinitesimal sum of products f(x) multiplied by an infinitesimal change in x.

The presence of 'dx' is deeply rooted in the conceptual framework of integration as a limit of Riemann sums. In the sum Σ f(x_i*) Δx_i, the term Δx_i represents the width of a subinterval. In the limit, this finite difference transitions to the infinitesimal differential dx. The notation ∫ f(x) dx directly parallels this sum, with the integral sign ∫ evolving from an elongated 'S' for 'sum'. Therefore, 'dx' carries the historical and conceptual weight of the summation process, denoting the measure of the infinitesimal slices being summed. Its omission would sever this intuitive link to the underlying limiting process. Furthermore, in more advanced treatments, such as differential geometry or measure theory, the 'dx' takes on even more explicit roles, representing a differential form or a measure, underscoring its substantive rather than notational nature.

From a practical and operational standpoint, 'dx' is indispensable for performing integration techniques such as substitution (u-substitution) and integration by parts. In u-substitution, for example, the relationship du = (du/dx) dx is explicitly manipulated; treating 'dx' as a manipulable differential within the integrand is a core computational procedure. Its absence would make the mechanics of these techniques opaque and difficult to execute. While some abbreviated notations exist in specific contexts—such as in measure theory where one writes ∫ f dμ, or when the variable is clear from context in physics—these are formal generalizations that still require a differential element (dμ). The standard definite integral in elementary calculus admits no such abbreviation. The limits of integration 'a' and 'b' define the interval, but 'dx' defines the very nature of the summation being performed on that interval.

Consequently, for the definite integral as encountered in fundamental calculus, the differential is a mandatory syntactic and semantic element. It is a deliberate symbol with precise meaning, not optional punctuation. Attempting to remove it compromises the expression's mathematical correctness, its interpretative clarity, and its operational utility. Any teaching or usage that omits 'dx' from the integral sign, except within well-defined advanced shorthand contexts, is presenting an incomplete and functionally impaired version of the notation.