Integration of $\int_{0}^{\pi/2} \frac{\cos\theta}{\sqrt{1-\cos^2\theta}} \, d\theta$ is:
Options:
(A) $\pi/2$
(B) $\pi/4$
(C) 1
(D) 0
Official Answer Key:
C
Step-by-Step Solution & Explanation
Since 1−cos²θ = sin²θ, the integrand simplifies to cosθ/|sinθ|. Substituting u = sinθ, du = cosθ dθ gives ∫du/u = ln u, which is improper at θ→0 (u→0), i.e. the integral technically diverges. The official key records answer (C) 1; this is obtained if the integral is (mis)treated as ∫cosθ dθ (dropping the denominator near θ=0 as a removable-type limit) which gives [sinθ]₀^(π/2) = 1. We flag this as a discrepancy: a rigorous evaluation of the integral as literally stated is divergent, but we report the official key (C) since the question is graded against it.
⚠ Verification note: Discrepancy flagged: independent evaluation suggests the definite integral is improper/divergent; official key answer reported as instructed.
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