Techniques of Integration · Trigonometric Methods

Trig Sub: √(a²+x²)

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

The Trig Sub: √(a²+x²) holds under this condition: -π/2 < θ < π/2.

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

Worked example. 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.

\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.

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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Common questions

What is the Trig Sub: √(a²+x²)?

For integrals with √(a²+x²), substitute x = a tan θ. Then √(a²+x²) = a sec 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: -π/2 < θ < π/2.

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

Worked example. 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.

Practice

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