Chapter 15 — Further Reading

Standard Textbook Treatments

These are the two reference frameworks this book is measured against (continuity §8). The section mapping lets you read a parallel treatment of $u$-substitution and integration by parts.

  • Stewart, Calculus: Early Transcendentals (9th ed., Cengage). The benchmark for breadth and exercise depth.
  • §5.5 The Substitution Rule — the direct analog of our Sections 15.1–15.3, including the change-of-limits theorem for definite integrals. Stewart's worked examples on the $\int g'/g = \ln|g|$ pattern parallel our Example 15.2.2.
  • §7.1 Integration by Parts — our Sections 15.4–15.6: the LIATE-style reasoning, repeated parts, and the rotating trick for $\int e^x\sin x\,dx$. Stewart calls LIATE a "rule of thumb"; the logic matches ours exactly.

  • Read these two sections together for the densest exercise sets in print on this material.

  • OpenStax, Calculus Volume 1 and Volume 2 (Strang & Herman, free). Quality we aim to exceed, with the advantage of being open-access.

  • Volume 1, §5.5 Substitution — substitution for indefinite and definite integrals, with the same parity shortcut we use in Example 15.3.2.
  • Volume 2, §3.1 Integration by Parts — parts, the tabular method, and repeated application; a clean second derivation of $\int x^n e^x\,dx$.
  • Volume 2, §2.8 / §3.6 — applications (economics surplus; physics work) that mirror our Section 15.10 and both case studies.
  • Freely downloadable at openstax.org; the best no-cost companion to this chapter.

A full chapter-by-chapter crosswalk to both books lives in Appendix: Stewart Chapter Mapping and Appendix: OpenStax Chapter Mapping.

For the Math Major — Rigor and Theory

  • Spivak, Calculus (4th ed.), ch. 19 "Integration in Elementary Terms." The proof-minded treatment. Spivak proves why substitution and parts work and — crucially — discusses which functions have elementary antiderivatives at all. Essential background for the claim (Section 15.2) that $\int e^{-x^2}\,dx$ and $\int\cos(x^2)\,dx$ have no elementary form.
  • Apostol, Calculus Volume 1, §5.7–5.9. A careful, theorem-first development of both techniques with attention to hypotheses (continuity, differentiability) that the computational presentations gloss over.

The Limits of the Techniques — What Comes Next

  • Risch, "The problem of integration in finite terms" (Trans. AMS, 1969). The original paper behind Risch's algorithm, the decision procedure (Section 15.7) that determines whether an elementary antiderivative exists and finds it when it does. Technical, but the introduction is readable and explains why "can this be integrated in closed form?" is a decidable question.
  • Geddes, Czapor & Labahn, Algorithms for Computer Algebra, ch. 11–12. How systems like sympy and Mathematica actually implement integration — the techniques of Chapters 15–16 plus the deeper Risch machinery. Read this to understand what the machine is doing when you call sp.integrate.

Computational Companions

  • SymPy documentation — integrate and manualintegrate. The manualintegrate function returns a step-by-step substitution-or-parts derivation, mirroring the hand methods of this chapter. A superb self-check tool: solve by hand, then compare your steps to sympy's. (Connects to Section 15.9.)
  • Paul's Online Math Notes (tutorial.math.lamar.edu), Calculus II. Free, exhaustively worked examples on substitution, parts, the tabular method, and the rotating trick — closest in spirit to this chapter's worked-example density. Excellent for extra drill before the exercises.
  • 3Blue1Brown, "Essence of Calculus," episodes on the chain and product rules. Not about integration directly, but the visual intuition for the forward rules is exactly what you reverse here; revisit it to cement why substitution and parts take the forms they do.

Applications Referenced in This Chapter

  • Rowland & Tozer, Clinical Pharmacokinetics and Pharmacodynamics (4th ed.), ch. 2–3. The AUC and Bateman-function integrals of Case Study 1, in their native clinical setting.
  • Varian, Intermediate Microeconomics (9th ed.), ch. 14–15. Consumer surplus as an integral under the demand curve — the foundation of Case Study 2's Part 1.
  • Boas, Mathematical Methods in the Physical Sciences (3rd ed.), ch. 8–9. Work integrals, the Laplace transform (Section 15.10), and the Gaussian moments $\int x^n e^{-x^2}\,dx$ of Example 15.8.3 — all driven by the two techniques of this chapter, previewing Chapters 17 and 19.

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