Techniques of Integration · Trigonometric Methods
Trig Sub: √(a²-x²)
For integrals with √(a²-x²), substitute x = a sin θ. Then √(a²-x²) = a cos θ.
- 2 worked examples
- 16 in Techniques of Integration
Worked examples
- x = 3sinθ, dx = 3cosθ dθ, √(9-9sin²θ) = 3cosθ
- ∫ 9cos²θ dθ = (9/2)∫ (1+cos2θ) dθ = (9/2)(θ + sin2θ/2) + C
- 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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Common questions
What is the Trig Sub: √(a²-x²)?
For integrals with √(a²-x²), substitute x = a sin θ. Then √(a²-x²) = a cos θ. It is one of the techniques of integration formulas in the CalcRef reference.
When does the Trig Sub: √(a²-x²) apply?
The Trig Sub: √(a²-x²) holds under this condition: -a ≤ x ≤ a, -π/2 ≤ θ ≤ π/2.
How do you use the Trig Sub: √(a²-x²)?
Worked example. Find ∫ √(9-x²) dx. x = 3sinθ, dx = 3cosθ dθ, √(9-9sin²θ) = 3cosθ. ∫ 9cos²θ dθ = (9/2)∫ (1+cos2θ) dθ = (9/2)(θ + sin2θ/2) + C. Back-substitute: (9/2)(arcsin(x/3) + x√(9-x²)/9) + C. Answer: (9/2)arcsin(x/3) + (x/2)√(9-x²) + C.
Practice
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Trig Sub: √(a²-x²) is one of 16 Techniques of Integration formulas in CalcRef, out of 136 in the library. Today's free round is 10 mixed multiple-choice questions drawn from the same dataset as this page, refreshed every day and replayable as often as you like - no account needed.