Book: Institutiones calculi integralis

Overview
Leonhard Euler's Institutiones calculi integralis presents an ambitious, systematic foundation for the theory and practice of integral calculus as understood in the late eighteenth century. The text gathers methods for finding antiderivatives, for evaluating definite integrals, and for reducing complex integrals to simpler forms, while weaving connections between integration, differential equations, infinite series, and special functions. The style is analytic and methodical, with attention to both technique and the underlying algebraic and transcendental structures that shape integrability.

Organization and Scope
The material unfolds in a deliberate progression from elementary to advanced topics. Early sections develop basic principles and standard procedures for integration, treating algebraic and rational functions, roots and binomials, and the classical substitutions that simplify integrands. Later portions address integrals of transcendental functions, series expansions, and the reduction of integrals involving radicals and trigonometric expressions to manageable canonical forms. The work also dedicates substantial attention to integrals that arise from solving differential equations, treating integration as both an end and a tool.

Core Techniques and Methods
Euler crystallizes a toolbox of integration techniques that includes substitution, decomposition into partial fractions, integration by parts and reduction formulas, and the systematic use of infinite series to represent and integrate otherwise intractable expressions. He emphasizes algebraic manipulation to transform integrals into standard patterns and develops general procedures for building reduction relations that lower the degree or complexity of integrands step by step. The use of analytic continuation and formal manipulations of series allows many integrals to be expressed in terms of previously studied quantities.

Differential Equations and Integral Relations
Integration and differential equations are treated as two facets of a single analytical enterprise. Euler shows how many integrals naturally emerge from first- and higher-order ordinary differential equations and how the process of finding integrals informs the solution of such equations. He exposes methods for finding integrating factors, reducing order, and employing substitutions that convert differential equations into quadratures. The systematic linkage of quadrature methods to differential equations helped establish a coherent framework for later work on analytic solutions.

Special Integrals and Functions
The Institutiones explores integrals that define or involve special functions, including logarithmic, exponential, and inverse trigonometric forms, and touches on classes of integrals that foreshadow elliptic and other transcendental integrals. Euler elucidates properties and transformations of these integrals, often expressing them via series or through functional relations that make evaluation and comparison feasible. His manipulations of infinite products and series, and his skill at reducing complicated expressions to known canonical integrals, laid groundwork for later formalization of special functions.

Impact and Legacy
The work became a cornerstone of mathematical analysis and a primary reference for generations of mathematicians. Its systematic approach influenced subsequent treatises on integral calculus and the teaching of analysis, shaping both technique and pedagogy. The blend of practical methods, theoretical insight, and algebraic ingenuity exemplified in these pages helped to elevate integration from a collection of ad hoc tricks to a coherent mathematical discipline, and echoes of its methods appear in modern integral tables, textbooks, and the broader development of analytic function theory.