Integrals with integrand functions fitting into the perimeter of those triangles are solvable by trigonometric substitution, like the exponential and hyperbolic integrals.
Inhaltsverzeichnis (Table of Contents)
- Introduction
- Properties of transcendental functions
- Properties of exponential and logarithmic functions
- Properties of trigonometric functions
- Properties of hyperbolic functions
- Table of Standard Integrals
- I. Rational powers of f(x)
- II. Exponential functions
- III. Trigonometric functions
- IV. Inverse trigonometric functions
- V. Hyperbolic functions
- VI. Inverse hyperbolic functions
- Trigonometric reduction formulas
Zielsetzung und Themenschwerpunkte (Objectives and Key Themes)
This essay aims to explore the significant role of right angle triangles in integral calculus. It focuses on how trigonometric substitution, particularly with functions that fit within the perimeter of right angle triangles, can solve integrals that involve exponential and hyperbolic functions.
- Trigonometric substitution for integral calculus
- Right angle triangles and their application to trigonometric functions
- Integration of exponential and hyperbolic functions using trigonometric substitution
- Properties of transcendental functions, including exponential, logarithmic, trigonometric, and hyperbolic functions
- Standard integral formulas for various types of functions
Zusammenfassung der Kapitel (Chapter Summaries)
- Introduction: This chapter introduces the concept of bypassing inverse functions through right angle triangles. It demonstrates the use of trigonometric functions (sine, cosine, tangent, etc.) and their respective inverse functions in solving integrals.
- Properties of transcendental functions: This section outlines the key properties of exponential, logarithmic, trigonometric, and hyperbolic functions. It explores relationships between different functions within each category, providing essential mathematical foundations for later sections.
- Table of Standard Integrals: This chapter presents a comprehensive list of standard integrals for various functions, including rational powers, exponential functions, trigonometric functions, inverse trigonometric functions, hyperbolic functions, and inverse hyperbolic functions. This table serves as a reference guide for integrating different types of functions.
- Trigonometric reduction formulas: This section discusses the application of trigonometric reduction formulas in simplifying high-power trigonometric integrals. It provides examples of reduction formulas for sine, cosine, tangent, cotangent, secant, and cosecant functions.
Schlüsselwörter (Keywords)
This essay focuses on the application of trigonometric substitution in integral calculus, utilizing right angle triangles to solve integrals with exponential and hyperbolic functions. Key terms include trigonometric functions, inverse trigonometric functions, hyperbolic functions, integration, trigonometric substitution, standard integrals, and reduction formulas.
- Quote paper
- Prof. Dr. med. John Bredakis (Author), 2011, The mighty role of right angle triangles to integral calculus, Munich, GRIN Verlag, https://www.hausarbeiten.de/document/170468
