Techniques of Integration · Trigonometric Methods

Trig Sub: √(x²-a²)

Q: When does the Trig Sub: √(x²-a²) apply?

The Trig Sub: √(x²-a²) holds under this condition: x ≥ a or x ≤ -a. 0 ≤ θ < π/2 or π ≤ θ < 3π/2.

Q: How do you use the Trig Sub: √(x²-a²)?

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

\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.

2 worked examples
16 in Techniques of Integration

Worked examples

Find ∫ 1/(x²√(x²-1)) dx.

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

Answer: √(x²-1)/x + C

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Find ∫ 1/√(x² - 4) dx.

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Common questions

What is the Trig Sub: √(x²-a²)?

For integrals with √(x²-a²), substitute x = a sec θ. Then √(x²-a²) = a tan θ. It is one of the techniques of integration formulas in the CalcRef reference.

When does the Trig Sub: √(x²-a²) apply?

The Trig Sub: √(x²-a²) holds under this condition: x ≥ a or x ≤ -a. 0 ≤ θ < π/2 or π ≤ θ < 3π/2.

How do you use the Trig Sub: √(x²-a²)?

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

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