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 (a2−x2)(\sqrt{a^2 - x^2}): x=asin⁡θ,dx=acos⁡θ dθ,a2−x2=acos⁡θx = a \sin\theta, \quad dx = a \cos\theta\, d\theta, \quad \sqrt{a^2 - x^2} = a \cos\theta

    • For (a2+x2)(\sqrt{a^2 + x^2}): x=atan⁡θ,dx=asec⁡2θ dθ,a2+x2=asec⁡θx = a \tan\theta, \quad dx = a \sec^2\theta\, d\theta, \quad \sqrt{a^2 + x^2} = a \sec\theta

    • For (x2−a2)(\sqrt{x^2 - a^2}): x=asec⁡θ,dx=asec⁡θtan⁡θ dθ,x2−a2=atan⁡θx = 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 xx and dxdx.

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

    4. Back-substitute to xx using trigonometric relationships:

      • If x=asin⁡θx = a\sin\theta: sin⁡θ=xa,cos⁡θ=a2−x2a,tan⁡θ=xa2−x2\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=atan⁡θx = a\tan\theta: tan⁡θ=xa,sec⁡θ=a2+x2a\tan\theta = \frac{x}{a}, \quad \sec\theta = \frac{\sqrt{a^2 + x^2}}{a}

      • If x=asec⁡θx = a\sec\theta: sec⁡θ=xa,cos⁡θ=ax,tan⁡θ=x2−a2a\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 xx.

  • Example: ∫dxa2−x2\int \frac{dx}{\sqrt{a^2 - x^2}}

    • Using x=asin⁡θx = a\sin\theta

    • Integral becomes: ∫acos⁡θ dθacos⁡θ=∫dθ=θ+C=arcsin⁡(xa)+C\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 xx.

    • [ ] Practice partial fraction setups.

    • [ ] Verify results by differentiation and domain checks.