Real integral by contour integration

Let I = ∫₀^2π dθ/(5 + 4 sin θ). Put z = e^(iθ) on |z| = 1: sin θ = (z − 1/z)/(2i), dθ = dz/(iz).

5 + 4 sin θ = 5 + (2/i)(z − 1/z) = 5 − 2iz + 2i/z
            = (−2i z² + 5z + 2i)/z
I = ∮_C [z/(−2i z² + 5z + 2i)] · dz/(iz)
  = ∮_C dz / [i(−2i z² + 5z + 2i)]
  = ∮_C dz / (2z² + 5iz − 2)

Factor: 2z² + 5iz − 2 = 2(z + 2i)(z + i/2). Poles: z = −2i (outside the unit circle) and z = −i/2 (inside).

Res at z = −i/2 = 1 / [2(z + 2i)] at z = −i/2 = 1/(2 · 3i/2) = 1/(3i)
I = 2πi × 1/(3i) = 2π/3

Answer: I = 2π/3. (Check: the standard formula 2π/√(a² − b²) = 2π/√(25 − 16) = 2π/3.)