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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.
About the Author
Special Substitutions
Euler Substitutions
First Euler substitution
Second Euler substitution
Third Euler substitution
Abel’s Substitution
The Differential Binomial and Chebyshev’s Theorem
Hyperbolic Substitutions for Integrals
General Trigonometric Substitution
Restrictions and extensions
Arithmetic-Geometric Mean and the Gauss Formula
The arithmetic-geometric mean
The Gauss formula
Some Interesting Examples
Solving Integrals by Differentiation with Respect to a Parameter
General Examples
Using Differential Equations
Advanced Techniques
The Basel Problem and Related Integrals
Special integrals with arctangents
Several integrals with logarithms
Some Theorems
Solving Logarithmic Integrals by Using Fourier Series
Examples
A Binet Type Formula for the Log-Gamma Function
Evaluating Integrals by Laplace and Fourier Transforms. Integrals Related to Riemann’s Zeta Function
Laplace Transform
A Tale of Two Integrals
Parseval’s Theorem
Some Important Hyperbolic Integrals
Expansion of the cotangent in partial fractions
Evaluation of important hyperbolic integrals
Exponential Polynomials and Gamma Integrals
The Functional Equation of the Riemann Zeta Function
The Functional Equation for Euler’s L(s) Function
Euler’s Formula for Zeta(2n)
Bernoulli numbers
Euler numbers and Euler’s formula for L(2n 1)
Various Techniques
The Formula of Poisson
Frullani Integrals
A Special Formula
Miscellaneous Selected Integrals
Catalan’s Constant
Summation of Series by Using Integrals
Generating Functions for Harmonic and Skew-Harmonic Numbers
Harmonic numbers
Skew-harmonic numbers
Double integrals related to the above series
Expansions of dilogarithms and trilogarithms
Fun with Lobachevsky
More Special Functions
Appendix A. List of Solved Integrals