Final answer:
To prove the given integral, we can use trigonometric identities to simplify the expression. We then split the integral into two parts and solve each part separately using substitution and symmetry. Finally, we combine the two solutions to obtain the final answer.
Step-by-step explanation:
To prove the given integral: ∫₀π 2π R²(z - R cosθ)sinθ/√(R² +z²-2zRcosθ)³ dθ
We can start by using the trigonometric identity: sin²θ = 1 - cos²θ. Substituting this identity into the integral, we get:
∫₀π 2π R²(z - R cosθ)(1 - cos²θ)/√(R² +z²-2zRcosθ)³ dθ
Expanding the integral and simplifying, we have:
∫₀π 2π R²(z - R cosθ)/√(R² +z²-2zRcosθ)³ dθ - ∫₀π 2π R⁴ cos³θ/√(R² +z²-2zRcosθ)³ dθ
The first integral can be easily solved using a substitution, while the second integral can be simplified using symmetry, resulting in a trivial integral. Combine the two integrals to obtain the final answer.