 Solution

ID: past-exam-of-the-mathematics-course-of-the-year-1-of-the-natural-sciences-course-of-the-university-of-cambridge/2005/paper-2/11/c/solution

Codex  0 Created 2026-10-18 Updated 2026-10-20

θ = π /4 - φ /2

, so

sin φ = cos 2 θ

and

d φ = - 2 d θ

. Reversing the transformed limits gives

\int_{0}^{π /2} \frac{d φ}{1 + sin φ} = 2 \int_{0}^{π /4} \frac{d θ}{1 + cos 2 θ} = 1.

The original integrand is finite at both endpoints.

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