CalcRef

Techniques of Integration · Trigonometric Methods

Trig Sub: √(a²-x²)

a2−x2: let x=asin⁡θ,  dx=acos⁡θ dθ\sqrt{a^2 - x^2}: \text{ let } x = a\sin\theta,\; dx = a\cos\theta\, d\theta

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

Conditions. -a ≤ x ≤ a, -π/2 ≤ θ ≤ π/2.

Worked examples

Find ∫ √(9-x²) dx.
  1. x = 3sinθ, dx = 3cosθ dθ, √(9-9sin²θ) = 3cosθ
  2. ∫ 9cos²θ dθ = (9/2)∫ (1+cos2θ) dθ = (9/2)(θ + sin2θ/2) + C
  3. Back-substitute: (9/2)(arcsin(x/3) + x√(9-x²)/9) + C

Answer: (9/2)arcsin(x/3) + (x/2)√(9-x²) + C

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