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Full Description
While differentiating elementary functions is merely a skill, finding their integrals is an art. This practical introduction to the art of integration gives readers the tools and confidence to tackle common and uncommon integrals. After a review of the basic properties of the Riemann integral, each chapter is devoted to a particular technique of elementary integration. Thorough explanations and plentiful worked examples prepare the reader for the extensive exercises at the end of each chapter. These exercises increase in difficulty from warm-up problems, through drill examples, to challenging extensions which illustrate such advanced topics as the irrationality of π and e, the solution of the Basel problem, Leibniz's series and Wallis's product. The author's accessible and engaging manner will appeal to a wide audience, including students, teachers and self-learners. The book can serve as a complete introduction to finding elementary integrals, or as a supplementary text for any beginning course in calculus.
Contents
1. The Riemann integral; 2. Basic properties of the definite integral - Part I; 3. Some basic standard forms; 4. Basic properties of the definite integral - Part II; 5. Standard forms; 6. Integration by substitution; 7. Integration by parts; 8. Trigonometric integrals; 9. Hyperbolic integrals; 10. Trigonometric and hyperbolic substitutions; 11. Integrating rational functions by partial fraction decomposition; 12. Six useful integrals; 13. Inverse hyperbolic functions and integrals leading to them; 14. Tangent half-angle substitution; 15. Further trigonometric integrals; 16. Further properties for definite integrals; 17. Integrating inverse functions; 18. Reduction formulae; 19. Some other special techniques and substitutions; 20. Improper integrals; 21. Two important improper integrals; Appendix A. Partial fractions; Appendix B. Answers to selected exercises; Index.
