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Calculus I / Techniques of Integration

Overview

The Fundamental Theorem of Calculus and the basic antidifferentiation table handle elementary functions directly. This topic extends that reach to composites, products, rational functions, and integrands that grow or blow up. The payoff in quantitative finance is direct: partial fractions factor bond-pricing denominators, integration by parts derives option payoff present values, improper integrals evaluate perpetuity formulas, and trigonometric substitution handles the square roots that appear in stochastic-volatility moment calculations.

By the end of this topic you should be able to:

  • integrate products using integration by parts (IBP) with the LIATE heuristic;
  • evaluate ∫sin⁡mxcos⁡nx dx\int \sin^m x\cos^n x\,dx and ∫tan⁡mxsec⁡nx dx\int \tan^m x\sec^n x\,dx by trig identities;
  • substitute x=asin⁡θx = a\sin\theta, x=atan⁡θx = a\tan\theta, or x=asec⁡θx = a\sec\theta to remove square-root factors;
  • decompose a proper rational function into partial fractions and integrate each term;
  • apply the Trapezoid, Midpoint, and Simpson's rules and bound their errors;
  • evaluate Type I and Type II improper integrals using limits;
  • choose between these techniques using a systematic decision tree.

In this lesson

  1. Contents
  2. The big picture

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