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Calculus I / Techniques of Integration

Integration Strategy

Differentiation is algorithmic: apply the chain, product, and sum rules mechanically to any composite expression and you always get an answer. Integration is not — there is no universal algorithm, and elementary antiderivatives do not always exist (e−x2e^{-x^2}, sin⁡(x2)\sin(x^2), and ln⁡ln⁡x\ln\ln x are standard examples). What exists instead is a decision tree: a prioritised sequence of tests that routes a given integrand to the right technique without trial-and-error. Mastering this tree is as important as knowing any individual technique; it is the skill that converts a student who can do trig integrals in isolation into one who can sit down with an arbitrary integral and make progress.
Learning objectives. By the end of this note you should be able to:
  • classify a given integrand as trig product, rational, product-for-IBP, radical, or none of the above;
  • execute the four-step decision tree before attempting any technique;
  • identify non-elementary integrands and know to switch to numerical methods;
  • solve any moderate-difficulty integration problem with a systematic approach.

In this lesson

  1. The standard integral table
  2. The four-step decision tree
  3. Decision tree in practice
  4. Pitfalls in technique selection

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