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Techniques of Integration · Substitution Methods

Trigonometric Substitution (Overview)

a2−x2:x=asin⁡θa2+x2:x=atan⁡θx2−a2:x=asec⁡θ\sqrt{a^2 - x^2}: x = a\sin\theta \quad \sqrt{a^2 + x^2}: x = a\tan\theta \quad \sqrt{x^2 - a^2}: x = a\sec\theta

Three standard trig substitutions for integrals involving square roots of quadratics. Each eliminates the radical using a Pythagorean identity.

Worked examples

Find ∫ 1/√(4-x²) dx.
  1. Use x = 2 sin θ, dx = 2 cos θ dθ
  2. √(4-4sin²θ) = 2cosθ
  3. ∫ 2cosθ/(2cosθ) dθ = ∫ dθ = θ + C
  4. θ = arcsin(x/2)

Answer: arcsin(x/2) + C

Related formulas

Trig Sub: √(a²-x²)Trig Sub: √(a²+x²)Trig Sub: √(x²-a²)

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