Trigonometric Substitution Overview

Overview

This lecture covers trigonometric substitution, a technique used to evaluate integrals involving square roots of quadratic expressions by relating them to right triangles and trigonometric identities.

When to Use Trigonometric Substitution

• Use trig substitution for integrals with square roots of the form √(a²–x²), √(a²+x²), or √(x²–a²).
• Standard substitution, integration by parts, or trig integrals won’t work for these forms.

Triangles and Trig Relationships

• Relate the given square root to the Pythagorean theorem and draw an associated right triangle.
• Assign triangle sides so that 'x' and constants fit the relationship, identifying if the square root is a leg or hypotenuse.
• Set up sides so that a trig function (sin, tan, or sec) gives x over the constant: x/a.

The Three Standard Substitutions

• For √(a²–x²):
  • Let x = a sin θ
  • dx = a cos θ dθ
  • Use identity: 1 – sin²θ = cos²θ
• For √(a²+x²):
  • Let x = a tan θ
  • dx = a sec²θ dθ
  • Use identity: 1 + tan²θ = sec²θ
• For √(x²–a²):
  • Let x = a sec θ
  • dx = a sec θ tan θ dθ
  • Use identity: sec²θ – 1 = tan²θ

Process for Trig Substitution

• Identify a substitution based on the form of the square root.
• Express all x’s and dx in terms of θ using the derivative of the substitution.
• Substitute and simplify until you get a trigonometric integral.
• Solve the integral, then use the original triangle to convert θ back to x.
• For definite integrals, change limits to θ after substitution to avoid needing back-substitution.

Completing the Square

• If the integrand does not initially match trig sub forms, complete the square to write quadratics as (something)² ± (something)².
• Substitute for new variables if needed (e.g., u = x + 1).

Examples and Techniques

• Use the triangle from your substitution to find all needed trig ratios for back-substitution.
• Always simplify algebraically before integrating.
• If the resulting trig integral is complicated, use known identities or switch to sines/cosines to try substitution.
• For complicated powers, factor out constants before using identities.

Key Terms & Definitions

• Trigonometric Substitution — Replacing x with a trig function to simplify integrals involving square roots.
• Hypotenuse — The longest side in a right triangle, opposite the right angle; relates to the sum of squared legs.
• Pythagorean Identity — Trig identities like 1 + tan²θ = sec²θ or 1 – sin²θ = cos²θ.
• Completing the Square — Rewriting a quadratic as a squared binomial plus/minus a constant.

Action Items / Next Steps

• Practice drawing triangles and setting up correct trigonometric relationships for substitution.
• Complete assigned homework problems from Section 7.3 using trig substitution.
• Review how and when to complete the square for non-standard quadratic expressions.
• Prepare for algebraic integration techniques in Section 7.4.