Integration Techniques: Trig Substitution & Partial Fractions

Trig Substitution - Overview

• Used to evaluate integrals involving square roots of quadratics by transforming them into trigonometric functions.

• Three key substitutions:

  • For (\sqrt{a^2 - x^2}): x = a \sin\theta, \quad dx = a \cos\theta\, d\theta, \quad \sqrt{a^2 - x^2} = a \cos\theta

  • For (\sqrt{a^2 + x^2}): x = a \tan\theta, \quad dx = a \sec^2\theta\, d\theta, \quad \sqrt{a^2 + x^2} = a \sec\theta

  • For (\sqrt{x^2 - a^2}): x = a \sec\theta, \quad dx = a \sec\theta \tan\theta\, d\theta, \quad \sqrt{x^2 - a^2} = a \tan\theta

• Workflow:

  1. Identify radical type.

  2. Substitute x and dx.

  3. Simplify and integrate in terms of (\theta).

  4. Back-substitute to x using trigonometric relationships:

     • If x = a\sin\theta: \sin\theta = \frac{x}{a}, \quad \cos\theta = \frac{\sqrt{a^2 - x^2}}{a}, \quad \tan\theta = \frac{x}{\sqrt{a^2 - x^2}}

     • If x = a\tan\theta: \tan\theta = \frac{x}{a}, \quad \sec\theta = \frac{\sqrt{a^2 + x^2}}{a}

     • If x = a\sec\theta: \sec\theta = \frac{x}{a}, \quad \cos\theta = \frac{a}{x}, \quad \tan\theta = \frac{\sqrt{x^2 - a^2}}{a}

  5. Simplify for final expression in x.

• Example: \int \frac{dx}{\sqrt{a^2 - x^2}}

  • Using x = a\sin\theta

  • Integral becomes: \int \frac{a\cos\theta\, d\theta}{a\cos\theta} = \int d\theta = \theta + C = \arcsin\left(\frac{x}{a}\right) + C

• Tips: Ensure domain is real; simplify completely; verify by differentiation.

• Relevance: Used in physics, engineering, and probability problems.

Partial Fractions - Overview

• Decomposes rational functions into simpler, integrable rational expressions.

• Key idea: Factor denominator to express as sum of simpler fractions.

• When to use: For proper fractions (numerator degree < denominator degree) with factorable denominators (linear and/or irreducible quadratics).

Review and Connections

• Both trig substitution and partial fractions are fundamental for integral evaluation.

• Study tips:

  • Memorize trig substitutions and back-substitution rules.

  • Practice partial fraction setups for distinct linear, repeated, and irreducible quadratic factors.

  • Differentiate results to check accuracy.

• Principles: Simplification via substitution; decomposition into simpler parts.

• Relevance: Solve problems in various fields; accuracy and verification are crucial.

• Quick study checklist:

  • [ ] Memorize trig substitution patterns.

  • [ ] Practice back-substitution to x.

  • [ ] Practice partial fraction setups.

  • [ ] Verify results by differentiation and domain checks.