This volume contains techniques of integration which are not found in standard calculus and advanced calculus books. It can be considered as a map to explore many classical approaches to evaluate integrals. It is intended for students and professionals who need to solve integrals or like to solve integrals and yearn to learn more about the various methods they could apply. Undergraduate and graduate students whose studies include mathematical analysis or mathematical physics will strongly benefit from this material. Mathematicians involved in research and teaching in areas related to calculus, advanced calculus and real analysis will find it invaluable. The volume contains numerous solved examples and problems for the reader. These examples can be used in classwork or for home assignments, as well as a supplement to student projects and student research.
Author(s): Khristo N Boyadzhiev
Edition: 1
Publisher: WSPC
Year: 2021
Language: English
Pages: 400
City: Singapore
Tags: Integration Techniques; Special Integration Techniques; Integration Methods; Integral Evaluation; Calculus
Contents
Preface
About the Author
1. Special Substitutions
1.1 Introduction
1.2 Euler Substitutions
1.2.1 First Euler substitution
1.2.2 Second Euler substitution
1.2.3 Third Euler substitution
1.3 Abel’s Substitution
1.4 The Differential Binomial and Chebyshev’s Theorem
1.5 Hyperbolic Substitutions for Integrals
1.6 General Trigonometric Substitution
1.6.1 Restrictions and extensions
1.7 Arithmetic-Geometric Mean and the Gauss Formula
1.7.1 The arithmetic-geometric mean
1.7.2 The Gauss formula
1.8 Some Interesting Examples
2. Solving Integrals by Differentiation with Respect to a Parameter
2.1 Introduction
2.2 General Examples
2.3 Using Differential Equations
2.4 Advanced Techniques
2.5 The Basel Problem and Related Integrals
2.5.1 Introduction
2.5.2 Special integrals with arctangents
2.5.3 Several integrals with logarithms
2.6 Some Theorems
3. Solving Logarithmic Integrals by Using Fourier Series
3.1 Introduction
3.2 Examples
3.3 A Binet Type Formula for the Log-Gamma Function
4. Evaluating Integrals by Laplace and Fourier Transforms. Integrals Related to Riemann’s Zeta Function
4.1 Introduction
4.2 Laplace Transform
4.3 A Tale of Two Integrals
4.4 Parseval’s Theorem
4.5 Some Important Hyperbolic Integrals
4.5.1 Expansion of the cotangent in partial fractions
4.5.2 Evaluation of important hyperbolic integrals
4.6 Exponential Polynomials and Gamma Integrals
4.7 The Functional Equation of the Riemann Zeta Function
4.8 The Functional Equation for Euler’s L(s) Function
4.9 Euler’s Formula for Zeta(2n)
4.9.1 Bernoulli numbers
4.9.2 Euler numbers and Euler’s formula for L(2n+1)
5. Various Techniques
5.1 The Formula of Poisson
5.2 Frullani Integrals
5.3 A Special Formula
5.4 Miscellaneous Selected Integrals
5.5 Catalan’s Constant
5.6 Summation of Series by Using Integrals
5.7 Generating Functions for Harmonic and Skew-Harmonic Numbers
5.7.1 Harmonic numbers
5.7.2 Skew-harmonic numbers
5.7.3 Double integrals related to the above series
5.7.4 Expansions of dilogarithms and trilogarithms
5.8 Fun with Lobachevsky
5.9 More Special Functions
Appendix A. List of Solved Integrals
References
Index
