Final answer:
To evaluate the integral, we can use the method of residues and close the contour by adding a circular arc in the complex plane. The singularity lies outside the contour, so the integral along the new contour is equal to the sum of the residues inside. By calculating the residue at the singularity, we find that the value of the integral is 4π/√3.
Step-by-step explanation:
To evaluate the integral ∫₀² (π² - sinθ)/(2 - cosθ) dθ, we can use the method of residues. First, note that the integrand has a singularity at cosθ = 2, which corresponds to θ = arccos(2). This singularity lies outside the contour of integration, so we can close the contour by adding a circular arc in the complex plane.
The integral along this new contour is equal to the sum of the residues inside the contour. Since the singularity at cosθ = 2 is a simple pole, the residue can be found as the limit of (π² - sinθ)/(2 - cosθ) as θ approaches arccos(2) from below. By substituting θ = arccos(2) - ε and taking the limit as ε approaches 0, we can calculate the residue mathematically.
The residue turns out to be 4π/√3. Therefore, the value of the integral ∫₀² (π² - sinθ)/(2 - cosθ) dθ is equal to 4π/√3.