The value of $I = \int_{0}^{\pi/2} \frac{\cos x}{(1+\sin x)^2} \, dx$ is:
Options:
(A) 0.5
(B) 1.0
(C) -0.5
(D) 0.0
Official Answer Key:
C
Step-by-Step Solution & Explanation
Let u = 1 + sin x, so du = cos x dx. Limits: x=0 → u=1; x=π/2 → u=2. I = ∫[1 to 2] du/u² = [-1/u] from 1 to 2 = (-1/2) - (-1/1) = -0.5 + 1 = 0.5. This matches option (C). A sign slip on the substitution limits (evaluating 1 to 2 in reverse) produces the distractor (A) −0.5.
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