Integrals with integrand functions fitting into the perimeter of those triangles are solvable by trigonometric substitution, like the exponential and hyperbolic integrals.
Table of Contents
1. Introduction: Bypassing the inverse functions by right angle triangles
2. Properties of transcendental functions
2.1 A. Properties of exponential and logarithmic functions
2.2 B. Properties of trigonometric functions
2.3 C. Properties of hyperbolic functions
3. Table of Standard Integrals
3.1 I. Rational powers of f(x)
3.2 II. Exponential functions
3.3 III. Trigonometric functions
3.4 IV. Inverse trigonometric functions
3.5 V. Hyperbolic functions
3.6 VI. Inverse hyperbolic functions
4. Trigonometric reduction formulas
5. A list of integrals solved by trigonometric substitution
6. Hyperbolic integrals solved by trigonometric substitution
6.1 Category I. Not exclusively solved by trigonometric substitution
6.2 Category II. Exclusively solved by trigonometric substitution
7. The number π by Archimedes
8. Algebraic trasformations applied to quadratic trinomial suitable for integration
9. A golden rule for integrals often not recognized
10. Trigonometric transformations suitable for integration
11. Trigonometric integrals solved by trigonometric substitution
12. That's why I am fond of Trigonometry
13. Elliptic integrals
14. The period T of the mathematical pendulum
15. Change of the variable in the mathematical pendulum
16. As an epilogue
Research Objectives and Topics
The primary objective of this essay is to demonstrate an alternative approach to solving complex integrals by utilizing the geometric properties of right-angle triangles to bypass traditional inverse trigonometric functions. The work provides a systematic mathematical framework to simplify integration processes across various transcendental, hyperbolic, and elliptic functions.
- Geometric derivation of inverse trigonometric substitutions
- Development of trigonometric reduction formulas for high-power integrals
- Application of trigonometric transformations to complex definite integrals
- Analysis of physical problems, such as the mathematical pendulum, using elliptic integrals
Excerpt from the Book
Introduction: Bypassing the inverse functions by right angle triangles:
Integrals with integrand functions fitting into the perimeter of those triangles are solvable by trigonometric substitution, like the exponential and hyperbolic integrals.
u = arc sin (x/a) | u = arc csc (x/a) | u = arc tan (x/a)
A=√(a^2 - x^2) | B=√(x^2 - a^2) | Γ=√(x^2 + a^2)
A=a.cosu | B=a.cotu | Γ=a.secu
x=a.sinu | x=a.cscu | x=a.tanu
dx=a.cosu.du | dx=-a.cscu.cotu.du | dx=a.sec^2u.du
d/dx arc sin (x/a) | d/dx arc csc (x/a) | d/dx arc tan (x/a)
du/dx = 1/a . secu = 1/A | du/dx = -1/a . sinu.tanu = -a/x.B | du/dx = -1/a . cos^2 u = a/Γ^2
Summary of Chapters
1. Introduction: Bypassing the inverse functions by right angle triangles: Provides the foundational geometric framework for using triangles to simplify integral substitutions.
2. Properties of transcendental functions: Outlines essential algebraic properties of exponential, logarithmic, trigonometric, and hyperbolic functions used throughout the text.
3. Table of Standard Integrals: Compiles a comprehensive list of integral forms ranging from rational powers to inverse hyperbolic functions.
4. Trigonometric reduction formulas: Presents systematic formulas for reducing integrals with high powers of trigonometric functions.
5. A list of integrals solved by trigonometric substitution: Displays practical applications of the triangle method for solving various integral forms.
6. Hyperbolic integrals solved by trigonometric substitution: Explores the connection between hyperbolic functions and trigonometric substitution techniques.
7. The number π by Archimedes: Discusses the geometric estimation of π and its relation to trigonometric identities.
8. Algebraic trasformations applied to quadratic trinomial suitable for integration: Details techniques for processing quadratic expressions to facilitate easier integration.
9. A golden rule for integrals often not recognized: Highlights the importance of identifying the first derivative within the integrand function.
10. Trigonometric transformations suitable for integration: Examines specific transformations, such as the multiplication of nominator and denominator by secant, to solve complex integrals.
11. Trigonometric integrals solved by trigonometric substitution: Analyzes the integration of product forms involving sine and cosine functions.
12. That's why I am fond of Trigonometry: Reflects on the elegance of geometry and trigonometry in solving algebraic problems.
13. Elliptic integrals: Addresses the application of beta functions and series expansion in solving elliptic integral forms.
14. The period T of the mathematical pendulum: Demonstrates the practical physical application of elliptic integrals in describing pendulum motion.
15. Change of the variable in the mathematical pendulum: Explains the mathematical process of variable substitution to arrive at standard elliptic integral forms.
16. As an epilogue: Concludes with the author’s philosophical view on mathematics and dedication to the work.
Keywords
Integral calculus, Trigonometric substitution, Right-angle triangles, Transcendental functions, Hyperbolic integrals, Elliptic integrals, Reduction formulas, Mathematical pendulum, Archimedes, Fourier series, Dirac delta, Orthogonal functions, Geometry, Trigonometry, Variable transformation
Frequently Asked Questions
What is the core focus of this work?
The work explores a methodology to solve integrals by using right-angle triangles to bypass the need for traditional inverse trigonometric functions.
Which mathematical areas are primarily covered?
The essay covers calculus, specifically integral calculus, involving transcendental, trigonometric, hyperbolic, and elliptic functions.
What is the author's primary research goal?
The goal is to provide a simplified, systematic way for students and mathematicians to grasp and solve complex integral problems efficiently.
What scientific methodology is applied?
The author applies geometric constructions of right-angle triangles as a substitution tool to transform and solve complex differential and integral expressions.
What topics are addressed in the main body?
The main body treats standard integrals, reduction formulas, Fourier series, elliptic integrals, and physical problems like the motion of a mathematical pendulum.
Which keywords define this document?
Key terms include integral calculus, trigonometric substitution, elliptic integrals, and geometric algebra.
How is the mathematical pendulum analyzed in this text?
The pendulum is treated as a case of a non-linear second-order differential equation, solved by reducing its energy-based relationship to an elliptic integral form.
What is the "golden rule" mentioned by the author?
The rule involves examining the integrand function to determine if it contains (or can be modified to contain) its first derivative, which allows for an immediate solution.
How does the author treat the Dirac delta function?
The author describes it as an isosceles triangle with extreme height and narrow base, discussing its properties in the context of continuous and discontinuous functions.
- 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
