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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.

  • 2 worked examples
  • 16 in Techniques of Integration

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

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Find ∫ 1/(x² + 9) dx.

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Related formulas

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

See all 16 Techniques of Integration formulas on one printable sheet - the same statements and conditions as these pages, typeset and laid out for paper.

Common questions

What is the Trigonometric Substitution (Overview)?

Three standard trig substitutions for integrals involving square roots of quadratics. Each eliminates the radical using a Pythagorean identity. It is one of the techniques of integration formulas in the CalcRef reference.

How do you use the Trigonometric Substitution (Overview)?

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

Practice

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