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Calculus I / Techniques of Integration

Trigonometric Substitution

Integrands containing a2−x2\sqrt{a^2 - x^2}, a2+x2\sqrt{a^2 + x^2}, or x2−a2\sqrt{x^2 - a^2} cannot be handled by the basic substitution rule because there is no useful u=f(x)u = f(x) whose derivative clears the square root. The fix is to substitute the old variable xx as a trigonometric function of a new variable θ\theta: this is the inverse substitution, and the three standard choices are x=asin⁡θx = a\sin\theta, x=atan⁡θx = a\tan\theta, and x=asec⁡θx = a\sec\theta respectively. Each exploits a Pythagorean identity to collapse the square root. The result is a trigonometric integral — which the preceding lesson handled — after which back-substitution returns the answer to xx.
Learning objectives. By the end of this note you should be able to:
  • identify which of the three substitutions applies to a given square-root expression;
  • execute the substitution, simplify using trig identities, and evaluate the resulting trig integral;
  • back-substitute to xx using a right-triangle diagram;
  • handle integrands that require completing the square before the standard substitution.

In this lesson

  1. The three standard substitutions
  2. Case 1: a2−x2\sqrt{a^2 - x^2}, substitute x=asin⁡θx = a\sin\theta
  3. Case 2: a2+x2\sqrt{a^2 + x^2}, substitute x=atan⁡θx = a\tan\theta
  4. Case 3: x2−a2\sqrt{x^2 - a^2}, substitute x=asec⁡θx = a\sec\theta
  5. Completing the square
  6. Finance context

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