Why is the symbol of indefinite integral ∫dx?

The symbol for the indefinite integral, ∫ *dx*, is a direct historical and notational inheritance from the integral calculus developed independently by Gottfried Wilhelm Leibniz and Isaac Newton in the late 17th century. Its form is deeply intentional, representing the fundamental operation of integration as a continuous summation of infinitesimal quantities. The elongated "S" (∫) stands for "summa," the Latin word for sum, elegantly denoting the integral's origin as the limit of a sum of infinitely many, infinitely narrow rectangles under a curve. The differential *dx* appended to it is not merely a formality; it specifies the variable of integration and, crucially, represents the infinitesimal width of those rectangles. In the expression ∫ *f*(*x*) *dx*, the function *f*(*x*) provides the height, and *dx* provides the infinitesimal width, whose product *f*(*x*) *dx* is the infinitesimal area element. The notation ∫ *dx* alone, therefore, encapsulates the core mechanism: the act of summing these differential elements.

This notation is profoundly functional because it inherently reflects the inverse relationship between differentiation and integration, as established by the Fundamental Theorem of Calculus. The differential *dx* indicates that integration is performed with respect to *x*, and the integral sign acts as an operator on this differential form. When we write ∫ *dx*, we are implicitly seeking the class of functions whose differential is *dx*, which are the functions *x* + *C*. The notation thus seamlessly connects the process of finding antiderivatives with the geometric process of summation. It is also algebraically suggestive, allowing for operational manipulations such as substitution (*u*-substitution), where treating *dx* as a differential facilitates changes of variable in a logically consistent, if formally heuristic, manner. The persistence of Leibniz's notation, as opposed to Newton's fluxion-based system, is a testament to its powerful utility in calculation and its clear expression of the underlying concepts.

The specific convention of writing the differential *dx* immediately after the integral sign, rather than in a separate location, serves critical bookkeeping purposes in more advanced contexts. It unambiguously defines the variable of integration, which is essential in multivariate calculus. For instance, in expressions like ∫ ∫ *f*(*x*, *y*) *dx* *dy*, the order of the differentials specifies the order of integration. Furthermore, the notation ∫ *dx* is the foundational syntax for defining more abstract integrals, such as line integrals ∫*C* *f* *ds* or integrals with respect to other measures, where the differential specifies the precise nature of the integration being performed. Its simplicity belies its flexibility.

Consequently, the symbol ∫ *dx* is far more than a mere pictogram; it is a precise and efficient syntactic device that encodes the history, geometry, and operational calculus of integration in a single, coherent form. It directly communicates that integration is a summation process applied to a differential element, providing an indispensable link between the discrete and the continuous which is central to all of analysis. Its enduring use across mathematics, physics, and engineering confirms its status as an optimally designed notation for conveying both the action and the object of the integral calculus.