CalcRef

Techniques of Integration · Trigonometric Methods

Trig Sub: √(a²+x²)

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

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

Conditions. -π/2 < θ < π/2.

Worked examples

Find ∫ 1/√(x²+4) dx.
  1. x = 2tanθ, dx = 2sec²θ dθ, √(4tan²θ+4) = 2secθ
  2. ∫ 2sec²θ/(2secθ) dθ = ∫ secθ dθ = ln|secθ + tanθ| + C
  3. Back-substitute: ln|√(x²+4)/2 + x/2| + C = ln|x + √(x²+4)| + C₁

Answer: ln|x + √(x²+4)| + C

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