Techniques of Integration · Trigonometric Methods
Trig Sub: √(a²+x²)
For integrals with √(a²+x²), substitute x = a tan θ. Then √(a²+x²) = a sec θ.
Conditions. -π/2 < θ < π/2.
- 0 variables
- 2 worked examples
- 16 in Techniques of Integration
Worked examples
Find ∫ 1/√(x²+4) dx.
- x = 2tanθ, dx = 2sec²θ dθ, √(4tan²θ+4) = 2secθ
- ∫ 2sec²θ/(2secθ) dθ = ∫ secθ dθ = ln|secθ + tanθ| + C
- Back-substitute: ln|√(x²+4)/2 + x/2| + C = ln|x + √(x²+4)| + C₁
Answer: ln|x + √(x²+4)| + C
1 more worked examplePremium
Find ∫ 1/(x² + 9)^(3/2) dx.
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Practice
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