CalcRef

Techniques of Integration · Trigonometric Methods

Trig Sub: √(x²-a²)

x2−a2: let x=asec⁡θ,  dx=asec⁡θtan⁡θ dθ\sqrt{x^2 - a^2}: \text{ let } x = a\sec\theta,\; dx = a\sec\theta\tan\theta\, d\theta

For integrals with √(x²-a²), substitute x = a sec θ. Then √(x²-a²) = a tan θ.

Conditions. x ≥ a or x ≤ -a. 0 ≤ θ < π/2 or π ≤ θ < 3π/2.

Worked examples

Find ∫ 1/(x²√(x²-1)) dx.
  1. x = secθ, dx = secθ tanθ dθ, √(sec²θ-1) = tanθ
  2. ∫ secθ tanθ/(sec²θ · tanθ) dθ = ∫ cosθ dθ = sinθ + C
  3. Back-substitute: sinθ = √(x²-1)/x

Answer: √(x²-1)/x + C

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