Lecture Notes: Trigonometric Substitution

Introduction

• Topic: Integration technique called trigonometric substitution.
• Purpose: Used to integrate square root functions, which can be notoriously difficult to integrate.
• Main Idea: Utilize trig integrals to simplify and solve integrals containing square root terms.

Key Trigonometric Substitutions

• Square Root Forms and Their Substitutions:
  1. (\sqrt{a^2 - x^2}) (\Rightarrow) Let (x = a \sin \theta)
  2. (\sqrt{a^2 + x^2}) (\Rightarrow) Let (x = a \tan \theta)
  3. (\sqrt{x^2 - a^2}) (\Rightarrow) Let (x = a \sec \theta)

Detailed Example: (\sqrt{a^2 - x^2})

• Substitution: Let (x = a \sin \theta).
• Differential Calculation:
  • (dx = a \cos \theta \ d\theta).
• Square Root Simplification:
  • (\sqrt{a^2 - x^2} = a \cos \theta) using the identity (1 - \sin^2 \theta = \cos^2 \theta).

Other Examples

• (\sqrt{a^2 + x^2}) Example:
  • Use (x = a \tan \theta).
  • Identity: (\sec^2 \theta = 1 + \tan^2 \theta).
  • Result: (\sqrt{a^2 + x^2}) becomes (a \sec \theta).
• (\sqrt{x^2 - a^2}) Example:
  • Use (x = a \sec \theta).
  • Identity: (\sec^2 \theta = 1 + \tan^2 \theta).
  • Result: (\sqrt{x^2 - a^2}) becomes (a \tan \theta).

Example Problem

• Integral to Solve: (\int \frac{1}{\sqrt{x^2 - 9}} , dx)
• Identify Square Root Term: (x^2 - 9)
• Choose Substitution:
  • (x = 3 \sec \theta)
  • (dx = 3 \sec \theta \tan \theta , d\theta)
  • Square root becomes (3 \tan \theta).
• Simplification:
  • Cancel terms to simplify integral to (\int \sec \theta , d\theta).
• Integration of Secant:
  • Result: (\ln |\sec \theta + \tan \theta| + C)
• Back-Substitute to x:
  • (\sec \theta = \frac{x}{3})
  • (\tan \theta = \frac{\sqrt{x^2 - 9}}{3})

Final Notes

• Practice: Trig substitution takes time and practice to master.
• Challenges: Involves multiple layers (substitution, trig identities, back-substitution).
• Encouragement: Don’t worry if it doesn’t click immediately; it’s one of the more challenging topics.
• Conclusion: More practice will follow in class.