Trigonometric Substitution Notes

Trigonometric Substitutions for Integrals

Trigonometric Substitution Overview

  • Trigonometric substitutions are techniques used in calculus to simplify integrals, particularly those involving square roots or certain polynomial expressions. They often convert integrals into trigonometric integrals, which are usually easier to solve.
  • Key expressions for substitution include forms such as (x = a \sin(\theta)), (x = a \tan(\theta)), and (x = a \sec(\theta)) based on the integrand's specific structure.

Example 1: Integral Transformation Using (x = \sec(5))

Given Integral:

  • The integrand includes the expression (\sqrt{1 + x^2}).

Substitution:

  • Set (x = \sec(5)).

Integral Transformation:

  • Upon substitution, the integral will be transformed into a trigonometric integral.
  • No evaluation of the integral is needed at this step.

Example 2: Trigonometric Substitution for (\int \frac{x^3}{49 - 2x^2} \ dx)

Part (a): Determine an Appropriate Trigonometric Substitution

  • The integrand contains the expression (\sqrt{72x^2}).
  1. Use (x = 7\sin(\theta)), for (-\frac{\pi}{2} \leq \theta \leq \frac{\pi}{2}).
  2. Use (x = 7\tan(\theta)), for (-\frac{\pi}{2} < \theta < \frac{\pi}{2}).
  3. Use (x = 7\sec(\theta)), for (0 \leq \theta < \frac{\pi}{2}) or (\frac{\pi}{2} \leq \theta < \pi).

Part (b): Apply the Substitution

  • Apply the chosen substitution to transform the integral into a trigonometric integral.
  • No evaluation of the integral is needed at this step.

Example 3: Another Trigonometric Substitution for (\int \sqrt{\cdots} \ dx)

Part (a): Determine an Appropriate Trigonometric Substitution

  • The integrand contains the expression (x^2 - 5).
  1. Use (x = \sqrt{5}\sin(\theta)), where (-\frac{\pi}{2} < \theta < \frac{\pi}{2}).
  2. Use (x = 5\tan(\theta)), where (-\frac{\pi}{2} < \theta < \frac{\pi}{2}).
  3. Use (x = \sqrt{5}\sec(\theta)), where (0 \leq \theta < \frac{\pi}{2}) or (\frac{\pi}{2} \leq \theta < \pi).

Part (b): Apply the Substitution

  • Apply the chosen substitution to transform the integral into a trigonometric integral.
  • No evaluation of the integral is needed at this step.