$I = \int_{0}^{\infty} \frac{dx}{(x^2+1)^2}$ has the value:
Options:
(A) 0.785
(B) 0.915
(C) 1.000
(D) 1.245
Official Answer Key:
A
Step-by-Step Solution & Explanation
Use the reduction/standard result ∫₀^∞ dx/(x²+a²)² = π/(4a³). Here a = 1, so I = π/4 = 0.7854 ≈ 0.785. (Cross-check by substitution x = tanθ: dx = sec²θ dθ, x²+1 = sec²θ, so I = ∫₀^{π/2} cos²θ dθ = π/4, confirming the result.) Answer: (A) 0.785.
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