Contour ∫dθ/(cosθ+2)

Method. For an integral over 0 to 2π of a function of cos θ, put z = e^{iθ}. Then the interval becomes the unit circle |z| = 1 and

cos θ = (z + 1/z)/2, dθ = dz/(iz).

Solution

cos θ + 2 = (z + 1/z)/2 + 2 = (z² + 4z + 1)/(2z)

I = ∮_C [2z/(z² + 4z + 1)] · dz/(iz)
  = (2/i) ∮_C dz/(z² + 4z + 1),   C: |z| = 1

Poles: z² + 4z + 1 = 0 gives z = −2 ± √3.

• z = α = −2 + √3 ≈ −0.268: inside C (|α| < 1)
• z = β = −2 − √3 ≈ −3.732: outside C

Residue at z = α (simple pole):

Res = 1/(α − β) = 1/(2√3)

By the residue theorem:

I = (2/i) · 2πi · 1/(2√3) = 2π/√3

Answer: ∫₀^{2π} dθ/(cos θ + 2) = 2π/√3 ≈ 3.628

(Check with the standard result ∫₀^{2π} dθ/(a + cos θ) = 2π/√(a² − 1) for a > 1; here a = 2.)